[{"data":1,"prerenderedAt":13431},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space":3},{"id":4,"title":5,"body":6,"description":12,"draft":13418,"enableComment":13419,"extension":13420,"image":1051,"meta":13421,"navigation":13419,"onday":13422,"path":13423,"seo":13424,"stem":13425,"summary":13426,"tags":13427,"__hash__":13430},"blog\u002Fblog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space.md","Dimension Curse on Zero and Positive Curvature Space",{"type":7,"value":8,"toc":13404},"minimark",[9,13,16,21,24,173,410,602,1031,1999,2229,3438,3557,3620,3990,3993,4494,4497,4913,4975,5251,5283,5288,5354,5513,5544,5796,5955,6101,6348,6351,6521,6640,6643,6934,6937,7127,7172,7420,7423,7867,7870,7975,8070,8073,8551,8554,8698,8701,8832,8834,9228,9262,9266,9343,9447,9450,9615,9618,9682,9684,9824,9827,9965,9969,9972,10004,10202,10205,10209,10212,10354,10593,10693,10941,10944,11336,11339,11396,11779,11782,11785,11788,11792,11795,11926,11929,12112,12114,12341,12344,12350,12353,12608,12612,12690,12851,12853,13032,13035,13176,13279,13283,13286,13289,13300,13303],[10,11,12],"p",{},"The curse of dimensionality is not a specific geometric structure, but a collection of phenomena induced by the intrinsic properties of high-dimensional spaces. In particular, effects such as concentration of measure and distance concentration lead to the degeneration of metric discriminability, making many distance-based methods ineffective.",[10,14,15],{},"This article introduces the Curse of Dimensionality effect in spaces of different curvature from mathematical and statistical perspectives.",[17,18,20],"h2",{"id":19},"euclidean-space","Euclidean Space",[10,22,23],{},"To introduce the curse of dimensionality, we need to start with the sample distribution in space.",[10,25,26,27,73,74,143,144,172],{},"Let us start with ",[28,29,33,55],"span",{"className":30,"translate":32},[31],"katex","no",[28,34,37],{"className":35},[36],"katex-mathml",[38,39,41],"math",{"xmlns":40},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[42,43,44,51],"semantics",{},[45,46,47],"mrow",{},[48,49,50],"mn",{},"1",[52,53,50],"annotation",{"encoding":54},"application\u002Fx-tex",[28,56,60],{"className":57,"ariaHidden":59},[58],"katex-html","true",[28,61,64,69],{"className":62},[63],"base",[28,65],{"className":66,"style":68},[67],"strut","height:0.6444em;",[28,70,50],{"className":71},[72],"mord","-dimension, a line segment in the numerical range ",[28,75,77,107],{"className":76,"translate":32},[31],[28,78,80],{"className":79},[36],[38,81,82],{"xmlns":40},[42,83,84,104],{},[45,85,86,91,94,96,99,101],{},[87,88,90],"mo",{"stretchy":89},"false","[",[87,92,93],{},"−",[48,95,50],{},[87,97,98],{"separator":59},",",[48,100,50],{},[87,102,103],{"stretchy":89},"]",[52,105,106],{"encoding":54},"[-1,1]",[28,108,110],{"className":109,"ariaHidden":59},[58],[28,111,113,117,121,124,127,131,136,139],{"className":112},[63],[28,114],{"className":115,"style":116},[67],"height:1em;vertical-align:-0.25em;",[28,118,90],{"className":119},[120],"mopen",[28,122,93],{"className":123},[72],[28,125,50],{"className":126},[72],[28,128,98],{"className":129},[130],"mpunct",[28,132],{"className":133,"style":135},[134],"mspace","margin-right:0.1667em;",[28,137,50],{"className":138},[72],[28,140,103],{"className":141},[142],"mclose",", and assume that all samples follow a random and uniform distribution on this line segment. Also, the unit measure of the line segment is ",[28,145,147,160],{"className":146,"translate":32},[31],[28,148,150],{"className":149},[36],[38,151,152],{"xmlns":40},[42,153,154,158],{},[45,155,156],{},[48,157,50],{},[52,159,50],{"encoding":54},[28,161,163],{"className":162,"ariaHidden":59},[58],[28,164,166,169],{"className":165},[63],[28,167],{"className":168,"style":68},[67],[28,170,50],{"className":171},[72]," unit of length.",[10,174,175,176,234,235,291,292,355,356,409],{},"Now, we consider the inner region that contains 90% of the measure, i.e., the interval ",[28,177,179,204],{"className":178,"translate":32},[31],[28,180,182],{"className":181},[36],[38,183,184],{"xmlns":40},[42,185,186,201],{},[45,187,188,190,192,195,197,199],{},[87,189,90],{"stretchy":89},[87,191,93],{},[48,193,194],{},"0.9",[87,196,98],{"separator":59},[48,198,194],{},[87,200,103],{"stretchy":89},[52,202,203],{"encoding":54},"[-0.9, 0.9]",[28,205,207],{"className":206,"ariaHidden":59},[58],[28,208,210,213,216,219,222,225,228,231],{"className":209},[63],[28,211],{"className":212,"style":116},[67],[28,214,90],{"className":215},[120],[28,217,93],{"className":218},[72],[28,220,194],{"className":221},[72],[28,223,98],{"className":224},[130],[28,226],{"className":227,"style":135},[134],[28,229,194],{"className":230},[72],[28,232,103],{"className":233},[142],", and treat the remaining 10% as a thin outer shell with finite thickness. It is evident that 90% of the samples are randomly distributed within ",[28,236,238,261],{"className":237,"translate":32},[31],[28,239,241],{"className":240},[36],[38,242,243],{"xmlns":40},[42,244,245,259],{},[45,246,247,249,251,253,255,257],{},[87,248,90],{"stretchy":89},[87,250,93],{},[48,252,194],{},[87,254,98],{"separator":59},[48,256,194],{},[87,258,103],{"stretchy":89},[52,260,203],{"encoding":54},[28,262,264],{"className":263,"ariaHidden":59},[58],[28,265,267,270,273,276,279,282,285,288],{"className":266},[63],[28,268],{"className":269,"style":116},[67],[28,271,90],{"className":272},[120],[28,274,93],{"className":275},[72],[28,277,194],{"className":278},[72],[28,280,98],{"className":281},[130],[28,283],{"className":284,"style":135},[134],[28,286,194],{"className":287},[72],[28,289,103],{"className":290},[142],", while the remaining 10% are located in the shell regions, namely ",[28,293,295,322],{"className":294,"translate":32},[31],[28,296,298],{"className":297},[36],[38,299,300],{"xmlns":40},[42,301,302,319],{},[45,303,304,306,308,310,312,314,316],{},[87,305,90],{"stretchy":89},[87,307,93],{},[48,309,50],{},[87,311,98],{"separator":59},[87,313,93],{},[48,315,194],{},[87,317,318],{"stretchy":89},")",[52,320,321],{"encoding":54},"[-1, -0.9)",[28,323,325],{"className":324,"ariaHidden":59},[58],[28,326,328,331,334,337,340,343,346,349,352],{"className":327},[63],[28,329],{"className":330,"style":116},[67],[28,332,90],{"className":333},[120],[28,335,93],{"className":336},[72],[28,338,50],{"className":339},[72],[28,341,98],{"className":342},[130],[28,344],{"className":345,"style":135},[134],[28,347,93],{"className":348},[72],[28,350,194],{"className":351},[72],[28,353,318],{"className":354},[142]," and ",[28,357,359,382],{"className":358,"translate":32},[31],[28,360,362],{"className":361},[36],[38,363,364],{"xmlns":40},[42,365,366,379],{},[45,367,368,371,373,375,377],{},[87,369,370],{"stretchy":89},"(",[48,372,194],{},[87,374,98],{"separator":59},[48,376,50],{},[87,378,103],{"stretchy":89},[52,380,381],{"encoding":54},"(0.9, 1]",[28,383,385],{"className":384,"ariaHidden":59},[58],[28,386,388,391,394,397,400,403,406],{"className":387},[63],[28,389],{"className":390,"style":116},[67],[28,392,370],{"className":393},[120],[28,395,194],{"className":396},[72],[28,398,98],{"className":399},[130],[28,401],{"className":402,"style":135},[134],[28,404,50],{"className":405},[72],[28,407,103],{"className":408},[142],".",[10,411,412,413,441,442,409],{},"Obviously, we can infer that the proportion of samples distributed on the thin shell in ",[28,414,416,429],{"className":415,"translate":32},[31],[28,417,419],{"className":418},[36],[38,420,421],{"xmlns":40},[42,422,423,427],{},[45,424,425],{},[48,426,50],{},[52,428,50],{"encoding":54},[28,430,432],{"className":431,"ariaHidden":59},[58],[28,433,435,438],{"className":434},[63],[28,436],{"className":437,"style":68},[67],[28,439,50],{"className":440},[72],"-dimension is ",[28,443,445,487],{"className":444,"translate":32},[31],[28,446,448],{"className":447},[36],[38,449,450],{"xmlns":40},[42,451,452,484],{},[45,453,454,475,477,480],{},[455,456,457,461,465],"msubsup",{},[458,459,460],"mi",{},"P",[462,463,464],"mtext",{},"shell",[45,466,467,470,473],{},[458,468,469],{},"d",[87,471,472],{},"=",[48,474,50],{},[87,476,472],{},[48,478,479],{},"10",[458,481,483],{"mathvariant":482},"normal","%",[52,485,486],{"encoding":54},"P_\\text{shell}^{d=1} = 10\\%",[28,488,490,591],{"className":489,"ariaHidden":59},[58],[28,491,493,497,581,585,588],{"className":492},[63],[28,494],{"className":495,"style":496},[67],"height:1.1322em;vertical-align:-0.2831em;",[28,498,500,505],{"className":499},[72],[28,501,460],{"className":502,"style":504},[72,503],"mathnormal","margin-right:0.13889em;",[28,506,509],{"className":507},[508],"msupsub",[28,510,514,572],{"className":511},[512,513],"vlist-t","vlist-t2",[28,515,518,567],{"className":516},[517],"vlist-r",[28,519,523,545],{"className":520,"style":522},[521],"vlist","height:0.8491em;",[28,524,526,531],{"style":525},"top:-2.4169em;margin-left:-0.1389em;margin-right:0.05em;",[28,527],{"className":528,"style":530},[529],"pstrut","height:2.7em;",[28,532,538],{"className":533},[534,535,536,537],"sizing","reset-size6","size3","mtight",[28,539,542],{"className":540},[72,541,537],"text",[28,543,464],{"className":544},[72,537],[28,546,548,551],{"style":547},"top:-3.063em;margin-right:0.05em;",[28,549],{"className":550,"style":530},[529],[28,552,554],{"className":553},[534,535,536,537],[28,555,557,560,564],{"className":556},[72,537],[28,558,469],{"className":559},[72,503,537],[28,561,472],{"className":562},[563,537],"mrel",[28,565,50],{"className":566},[72,537],[28,568,571],{"className":569},[570],"vlist-s","​",[28,573,575],{"className":574},[517],[28,576,579],{"className":577,"style":578},[521],"height:0.2831em;",[28,580],{},[28,582],{"className":583,"style":584},[134],"margin-right:0.2778em;",[28,586,472],{"className":587},[563],[28,589],{"className":590,"style":584},[134],[28,592,594,598],{"className":593},[63],[28,595],{"className":596,"style":597},[67],"height:0.8056em;vertical-align:-0.0556em;",[28,599,601],{"className":600},[72],"10%",[10,603,604,605,634,635,730,731,872,873,1030],{},"Now, let us extend this similar situation to circles in a ",[28,606,608,622],{"className":607,"translate":32},[31],[28,609,611],{"className":610},[36],[38,612,613],{"xmlns":40},[42,614,615,620],{},[45,616,617],{},[48,618,619],{},"2",[52,621,619],{"encoding":54},[28,623,625],{"className":624,"ariaHidden":59},[58],[28,626,628,631],{"className":627},[63],[28,629],{"className":630,"style":68},[67],[28,632,619],{"className":633},[72],"-dimensional plane. As we all know, the area of a circle can be calculated with the formula ",[28,636,638,666],{"className":637,"translate":32},[31],[28,639,641],{"className":640},[36],[38,642,643],{"xmlns":40},[42,644,645,663],{},[45,646,647,650,652,655],{},[458,648,649],{},"S",[87,651,472],{},[458,653,654],{},"π",[656,657,658,661],"msup",{},[458,659,660],{},"r",[48,662,619],{},[52,664,665],{"encoding":54},"S = \\pi r^2",[28,667,669,689],{"className":668,"ariaHidden":59},[58],[28,670,672,676,680,683,686],{"className":671},[63],[28,673],{"className":674,"style":675},[67],"height:0.6833em;",[28,677,649],{"className":678,"style":679},[72,503],"margin-right:0.05764em;",[28,681],{"className":682,"style":584},[134],[28,684,472],{"className":685},[563],[28,687],{"className":688,"style":584},[134],[28,690,692,696,700],{"className":691},[63],[28,693],{"className":694,"style":695},[67],"height:0.8141em;",[28,697,654],{"className":698,"style":699},[72,503],"margin-right:0.03588em;",[28,701,703,707],{"className":702},[72],[28,704,660],{"className":705,"style":706},[72,503],"margin-right:0.02778em;",[28,708,710],{"className":709},[508],[28,711,713],{"className":712},[512],[28,714,716],{"className":715},[517],[28,717,719],{"className":718,"style":695},[521],[28,720,721,724],{"style":547},[28,722],{"className":723,"style":530},[529],[28,725,727],{"className":726},[534,535,536,537],[28,728,619],{"className":729},[72,537],". Also, we consider the inner region that contains 90% of the measure, i.e., ",[28,732,734,768],{"className":733,"translate":32},[31],[28,735,737],{"className":736},[36],[38,738,739],{"xmlns":40},[42,740,741,765],{},[45,742,743,750,753,760,763],{},[656,744,745,748],{},[458,746,747],{},"x",[48,749,619],{},[87,751,752],{},"+",[656,754,755,758],{},[458,756,757],{},"y",[48,759,619],{},[87,761,762],{},"≤",[48,764,194],{},[52,766,767],{"encoding":54},"x^2 + y^2 \\leq 0.9",[28,769,771,818,863],{"className":770,"ariaHidden":59},[58],[28,772,774,778,807,811,815],{"className":773},[63],[28,775],{"className":776,"style":777},[67],"height:0.8974em;vertical-align:-0.0833em;",[28,779,781,784],{"className":780},[72],[28,782,747],{"className":783},[72,503],[28,785,787],{"className":786},[508],[28,788,790],{"className":789},[512],[28,791,793],{"className":792},[517],[28,794,796],{"className":795,"style":695},[521],[28,797,798,801],{"style":547},[28,799],{"className":800,"style":530},[529],[28,802,804],{"className":803},[534,535,536,537],[28,805,619],{"className":806},[72,537],[28,808],{"className":809,"style":810},[134],"margin-right:0.2222em;",[28,812,752],{"className":813},[814],"mbin",[28,816],{"className":817,"style":810},[134],[28,819,821,825,854,857,860],{"className":820},[63],[28,822],{"className":823,"style":824},[67],"height:1.0085em;vertical-align:-0.1944em;",[28,826,828,831],{"className":827},[72],[28,829,757],{"className":830,"style":699},[72,503],[28,832,834],{"className":833},[508],[28,835,837],{"className":836},[512],[28,838,840],{"className":839},[517],[28,841,843],{"className":842,"style":695},[521],[28,844,845,848],{"style":547},[28,846],{"className":847,"style":530},[529],[28,849,851],{"className":850},[534,535,536,537],[28,852,619],{"className":853},[72,537],[28,855],{"className":856,"style":584},[134],[28,858,762],{"className":859},[563],[28,861],{"className":862,"style":584},[134],[28,864,866,869],{"className":865},[63],[28,867],{"className":868,"style":68},[67],[28,870,194],{"className":871},[72],", and the remaining finite shell ",[28,874,876,911],{"className":875,"translate":32},[31],[28,877,879],{"className":878},[36],[38,880,881],{"xmlns":40},[42,882,883,908],{},[45,884,885,887,890,896,898,904,906],{},[48,886,194],{},[87,888,889],{},"\u003C",[656,891,892,894],{},[458,893,747],{},[48,895,619],{},[87,897,752],{},[656,899,900,902],{},[458,901,757],{},[48,903,619],{},[87,905,762],{},[48,907,50],{},[52,909,910],{"encoding":54},"0.9 \u003C x^2 + y^2 \\leq 1",[28,912,914,933,977,1021],{"className":913,"ariaHidden":59},[58],[28,915,917,921,924,927,930],{"className":916},[63],[28,918],{"className":919,"style":920},[67],"height:0.6835em;vertical-align:-0.0391em;",[28,922,194],{"className":923},[72],[28,925],{"className":926,"style":584},[134],[28,928,889],{"className":929},[563],[28,931],{"className":932,"style":584},[134],[28,934,936,939,968,971,974],{"className":935},[63],[28,937],{"className":938,"style":777},[67],[28,940,942,945],{"className":941},[72],[28,943,747],{"className":944},[72,503],[28,946,948],{"className":947},[508],[28,949,951],{"className":950},[512],[28,952,954],{"className":953},[517],[28,955,957],{"className":956,"style":695},[521],[28,958,959,962],{"style":547},[28,960],{"className":961,"style":530},[529],[28,963,965],{"className":964},[534,535,536,537],[28,966,619],{"className":967},[72,537],[28,969],{"className":970,"style":810},[134],[28,972,752],{"className":973},[814],[28,975],{"className":976,"style":810},[134],[28,978,980,983,1012,1015,1018],{"className":979},[63],[28,981],{"className":982,"style":824},[67],[28,984,986,989],{"className":985},[72],[28,987,757],{"className":988,"style":699},[72,503],[28,990,992],{"className":991},[508],[28,993,995],{"className":994},[512],[28,996,998],{"className":997},[517],[28,999,1001],{"className":1000,"style":695},[521],[28,1002,1003,1006],{"style":547},[28,1004],{"className":1005,"style":530},[529],[28,1007,1009],{"className":1008},[534,535,536,537],[28,1010,619],{"className":1011},[72,537],[28,1013],{"className":1014,"style":584},[134],[28,1016,762],{"className":1017},[563],[28,1019],{"className":1020,"style":584},[134],[28,1022,1024,1027],{"className":1023},[63],[28,1025],{"className":1026,"style":68},[67],[28,1028,50],{"className":1029},[72],". Next, we need to calculate the proportion of samples distributed in these two regions.",[28,1032,1035],{"className":1033,"translate":32},[1034],"katex-display",[28,1036,1038,1223],{"className":1037,"translate":32},[31],[28,1039,1041],{"className":1040},[36],[38,1042,1044],{"xmlns":40,"display":1043},"block",[42,1045,1046,1220],{},[1047,1048,1052,1096,1162],"mtable",{"rowspacing":1049,"columnalign":1050,"columnspacing":1051},"0.25em","right","",[1053,1054,1055],"mtr",{},[1056,1057,1058],"mtd",{},[1059,1060,1062],"mstyle",{"scriptlevel":1061,"displaystyle":59},"0",[45,1063,1064,1072,1074,1076,1079,1081,1083,1089,1091,1094],{},[1065,1066,1067,1069],"msub",{},[458,1068,649],{},[462,1070,1071],{},"inner",[87,1073,472],{},[458,1075,654],{},[87,1077,1078],{},"⋅",[87,1080,370],{"stretchy":89},[48,1082,194],{},[656,1084,1085,1087],{},[87,1086,318],{"stretchy":89},[48,1088,619],{},[87,1090,472],{},[48,1092,1093],{},"0.81",[458,1095,654],{},[1053,1097,1098],{},[1056,1099,1100],{},[1059,1101,1102],{"scriptlevel":1061,"displaystyle":59},[45,1103,1104,1110,1112,1119,1121,1127,1129,1131,1133,1139,1141,1143,1145,1147,1149,1155,1157,1160],{},[1065,1105,1106,1108],{},[458,1107,649],{},[462,1109,464],{},[87,1111,472],{},[1065,1113,1114,1116],{},[458,1115,649],{},[462,1117,1118],{},"unit",[87,1120,93],{},[1065,1122,1123,1125],{},[458,1124,649],{},[462,1126,1071],{},[87,1128,472],{},[458,1130,654],{},[87,1132,1078],{},[656,1134,1135,1137],{},[48,1136,50],{},[48,1138,619],{},[87,1140,93],{},[458,1142,654],{},[87,1144,1078],{},[87,1146,370],{"stretchy":89},[48,1148,194],{},[656,1150,1151,1153],{},[87,1152,318],{"stretchy":89},[48,1154,619],{},[87,1156,472],{},[48,1158,1159],{},"0.19",[458,1161,654],{},[1053,1163,1164],{},[1056,1165,1166],{},[1059,1167,1168],{"scriptlevel":1061,"displaystyle":59},[45,1169,1170,1184,1186,1201,1203,1213,1215,1218],{},[455,1171,1172,1174,1176],{},[458,1173,460],{},[462,1175,464],{},[45,1177,1178,1180,1182],{},[458,1179,469],{},[87,1181,472],{},[48,1183,619],{},[87,1185,472],{},[1187,1188,1189,1195],"mfrac",{},[1065,1190,1191,1193],{},[458,1192,649],{},[462,1194,464],{},[1065,1196,1197,1199],{},[458,1198,649],{},[462,1200,1118],{},[87,1202,472],{},[1187,1204,1205,1211],{},[45,1206,1207,1209],{},[48,1208,1159],{},[458,1210,654],{},[458,1212,654],{},[87,1214,472],{},[48,1216,1217],{},"19",[458,1219,483],{"mathvariant":482},[52,1221,1222],{"encoding":54},"\\begin{aligned}\nS_\\text{inner} = \\pi \\cdot (0.9)^2 = 0.81 \\pi \\\\\nS_\\text{shell} = S_\\text{unit} - S_\\text{inner} = \\pi \\cdot 1^2 - \\pi \\cdot (0.9)^{2} = 0.19\\pi \\\\\nP^{d=2}_\\text{shell} = \\frac{S_\\text{shell}}{S_\\text{unit}} = \\frac{0.19 \\pi}{\\pi} = 19\\%  \n\\end{aligned}",[28,1224,1226],{"className":1225,"ariaHidden":59},[58],[28,1227,1229,1233],{"className":1228},[63],[28,1230],{"className":1231,"style":1232},[67],"height:5.5445em;vertical-align:-2.5223em;",[28,1234,1236],{"className":1235},[72],[28,1237,1239],{"className":1238},[1047],[28,1240,1243],{"className":1241},[1242],"col-align-r",[28,1244,1246,1990],{"className":1245},[512,513],[28,1247,1249,1987],{"className":1248},[517],[28,1250,1253,1382,1663],{"className":1251,"style":1252},[521],"height:3.0223em;",[28,1254,1256,1260],{"style":1255},"top:-5.5185em;",[28,1257],{"className":1258,"style":1259},[529],"height:3.3603em;",[28,1261,1263,1309,1312,1315,1318,1321,1324,1327,1330,1333,1336,1367,1370,1373,1376,1379],{"className":1262},[72],[28,1264,1266,1269],{"className":1265},[72],[28,1267,649],{"className":1268,"style":679},[72,503],[28,1270,1272],{"className":1271},[508],[28,1273,1275,1300],{"className":1274},[512,513],[28,1276,1278,1297],{"className":1277},[517],[28,1279,1282],{"className":1280,"style":1281},[521],"height:0.3175em;",[28,1283,1285,1288],{"style":1284},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[28,1286],{"className":1287,"style":530},[529],[28,1289,1291],{"className":1290},[534,535,536,537],[28,1292,1294],{"className":1293},[72,541,537],[28,1295,1071],{"className":1296},[72,537],[28,1298,571],{"className":1299},[570],[28,1301,1303],{"className":1302},[517],[28,1304,1307],{"className":1305,"style":1306},[521],"height:0.15em;",[28,1308],{},[28,1310],{"className":1311,"style":584},[134],[28,1313,472],{"className":1314},[563],[28,1316],{"className":1317,"style":584},[134],[28,1319,654],{"className":1320,"style":699},[72,503],[28,1322],{"className":1323,"style":810},[134],[28,1325,1078],{"className":1326},[814],[28,1328],{"className":1329,"style":810},[134],[28,1331,370],{"className":1332},[120],[28,1334,194],{"className":1335},[72],[28,1337,1339,1342],{"className":1338},[142],[28,1340,318],{"className":1341},[142],[28,1343,1345],{"className":1344},[508],[28,1346,1348],{"className":1347},[512],[28,1349,1351],{"className":1350},[517],[28,1352,1355],{"className":1353,"style":1354},[521],"height:0.8641em;",[28,1356,1358,1361],{"style":1357},"top:-3.113em;margin-right:0.05em;",[28,1359],{"className":1360,"style":530},[529],[28,1362,1364],{"className":1363},[534,535,536,537],[28,1365,619],{"className":1366},[72,537],[28,1368],{"className":1369,"style":584},[134],[28,1371,472],{"className":1372},[563],[28,1374],{"className":1375,"style":584},[134],[28,1377,1093],{"className":1378},[72],[28,1380,654],{"className":1381,"style":699},[72,503],[28,1383,1385,1388],{"style":1384},"top:-3.9944em;",[28,1386],{"className":1387,"style":1259},[529],[28,1389,1391,1435,1438,1441,1444,1487,1490,1493,1496,1539,1542,1545,1548,1551,1554,1557,1560,1589,1592,1595,1598,1601,1604,1607,1610,1613,1616,1648,1651,1654,1657,1660],{"className":1390},[72],[28,1392,1394,1397],{"className":1393},[72],[28,1395,649],{"className":1396,"style":679},[72,503],[28,1398,1400],{"className":1399},[508],[28,1401,1403,1427],{"className":1402},[512,513],[28,1404,1406,1424],{"className":1405},[517],[28,1407,1410],{"className":1408,"style":1409},[521],"height:0.3361em;",[28,1411,1412,1415],{"style":1284},[28,1413],{"className":1414,"style":530},[529],[28,1416,1418],{"className":1417},[534,535,536,537],[28,1419,1421],{"className":1420},[72,541,537],[28,1422,464],{"className":1423},[72,537],[28,1425,571],{"className":1426},[570],[28,1428,1430],{"className":1429},[517],[28,1431,1433],{"className":1432,"style":1306},[521],[28,1434],{},[28,1436],{"className":1437,"style":584},[134],[28,1439,472],{"className":1440},[563],[28,1442],{"className":1443,"style":584},[134],[28,1445,1447,1450],{"className":1446},[72],[28,1448,649],{"className":1449,"style":679},[72,503],[28,1451,1453],{"className":1452},[508],[28,1454,1456,1479],{"className":1455},[512,513],[28,1457,1459,1476],{"className":1458},[517],[28,1460,1462],{"className":1461,"style":1281},[521],[28,1463,1464,1467],{"style":1284},[28,1465],{"className":1466,"style":530},[529],[28,1468,1470],{"className":1469},[534,535,536,537],[28,1471,1473],{"className":1472},[72,541,537],[28,1474,1118],{"className":1475},[72,537],[28,1477,571],{"className":1478},[570],[28,1480,1482],{"className":1481},[517],[28,1483,1485],{"className":1484,"style":1306},[521],[28,1486],{},[28,1488],{"className":1489,"style":810},[134],[28,1491,93],{"className":1492},[814],[28,1494],{"className":1495,"style":810},[134],[28,1497,1499,1502],{"className":1498},[72],[28,1500,649],{"className":1501,"style":679},[72,503],[28,1503,1505],{"className":1504},[508],[28,1506,1508,1531],{"className":1507},[512,513],[28,1509,1511,1528],{"className":1510},[517],[28,1512,1514],{"className":1513,"style":1281},[521],[28,1515,1516,1519],{"style":1284},[28,1517],{"className":1518,"style":530},[529],[28,1520,1522],{"className":1521},[534,535,536,537],[28,1523,1525],{"className":1524},[72,541,537],[28,1526,1071],{"className":1527},[72,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we can continue our exploration in ",[28,2003,2005,2019],{"className":2004,"translate":32},[31],[28,2006,2008],{"className":2007},[36],[38,2009,2010],{"xmlns":40},[42,2011,2012,2017],{},[45,2013,2014],{},[48,2015,2016],{},"3",[52,2018,2016],{"encoding":54},[28,2020,2022],{"className":2021,"ariaHidden":59},[58],[28,2023,2025,2028],{"className":2024},[63],[28,2026],{"className":2027,"style":68},[67],[28,2029,2016],{"className":2030},[72],"-dimensional space. The volume for a ",[28,2033,2035,2048],{"className":2034,"translate":32},[31],[28,2036,2038],{"className":2037},[36],[38,2039,2040],{"xmlns":40},[42,2041,2042,2046],{},[45,2043,2044],{},[48,2045,2016],{},[52,2047,2016],{"encoding":54},[28,2049,2051],{"className":2050,"ariaHidden":59},[58],[28,2052,2054,2057],{"className":2053},[63],[28,2055],{"className":2056,"style":68},[67],[28,2058,2016],{"className":2059},[72],"-dimensional sphere is ",[28,2062,2064,2096],{"className":2063,"translate":32},[31],[28,2065,2067],{"className":2066},[36],[38,2068,2069],{"xmlns":40},[42,2070,2071,2093],{},[45,2072,2073,2076,2078,2085,2087],{},[458,2074,2075],{},"V",[87,2077,472],{},[1187,2079,2080,2083],{},[48,2081,2082],{},"4",[48,2084,2016],{},[458,2086,654],{},[656,2088,2089,2091],{},[458,2090,660],{},[48,2092,2016],{},[52,2094,2095],{"encoding":54},"V=\\frac{4}{3} \\pi r^3",[28,2097,2099,2118],{"className":2098,"ariaHidden":59},[58],[28,2100,2102,2105,2109,2112,2115],{"className":2101},[63],[28,2103],{"className":2104,"style":675},[67],[28,2106,2075],{"className":2107,"style":2108},[72,503],"margin-right:0.22222em;",[28,2110],{"className":2111,"style":584},[134],[28,2113,472],{"className":2114},[563],[28,2116],{"className":2117,"style":584},[134],[28,2119,2121,2125,2197,2200],{"className":2120},[63],[28,2122],{"className":2123,"style":2124},[67],"height:1.1901em;vertical-align:-0.345em;",[28,2126,2128,2131,2194],{"className":2127},[72],[28,2129],{"className":2130},[120,1753],[28,2132,2134],{"className":2133},[1187],[28,2135,2137,2185],{"className":2136},[512,513],[28,2138,2140,2182],{"className":2139},[517],[28,2141,2144,2159,2167],{"className":2142,"style":2143},[521],"height:0.8451em;",[28,2145,2147,2150],{"style":2146},"top:-2.655em;",[28,2148],{"className":2149,"style":1773},[529],[28,2151,2153],{"className":2152},[534,535,536,537],[28,2154,2156],{"className":2155},[72,537],[28,2157,2016],{"className":2158},[72,537],[28,2160,2161,2164],{"style":1822},[28,2162],{"className":2163,"style":1773},[529],[28,2165],{"className":2166,"style":1830},[1829],[28,2168,2170,2173],{"style":2169},"top:-3.394em;",[28,2171],{"className":2172,"style":1773},[529],[28,2174,2176],{"className":2175},[534,535,536,537],[28,2177,2179],{"className":2178},[72,537],[28,2180,2082],{"className":2181},[72,537],[28,2183,571],{"className":2184},[570],[28,2186,2188],{"className":2187},[517],[28,2189,2192],{"className":2190,"style":2191},[521],"height:0.345em;",[28,2193],{},[28,2195],{"className":2196},[142,1753],[28,2198,654],{"className":2199,"style":699},[72,503],[28,2201,2203,2206],{"className":2202},[72],[28,2204,660],{"className":2205,"style":706},[72,503],[28,2207,2209],{"className":2208},[508],[28,2210,2212],{"className":2211},[512],[28,2213,2215],{"className":2214},[517],[28,2216,2218],{"className":2217,"style":695},[521],[28,2219,2220,2223],{"style":547},[28,2221],{"className":2222,"style":530},[529],[28,2224,2226],{"className":2225},[534,535,536,537],[28,2227,2016],{"className":2228},[72,537],[28,2230,2232],{"className":2231,"translate":32},[1034],[28,2233,2235,2429],{"className":2234,"translate":32},[31],[28,2236,2238],{"className":2237},[36],[38,2239,2240],{"xmlns":40,"display":1043},[42,2241,2242,2426],{},[1047,2243,2244,2296,2381],{"rowspacing":1049,"columnalign":1050,"columnspacing":1051},[1053,2245,2246],{},[1056,2247,2248],{},[1059,2249,2250],{"scriptlevel":1061,"displaystyle":59},[45,2251,2252,2258,2260,2266,2268,2270,2272,2274,2280,2282,2288,2290,2293],{},[1065,2253,2254,2256],{},[458,2255,2075],{},[462,2257,1071],{},[87,2259,472],{},[1187,2261,2262,2264],{},[48,2263,2082],{},[48,2265,2016],{},[458,2267,654],{},[87,2269,1078],{},[87,2271,370],{"stretchy":89},[48,2273,194],{},[656,2275,2276,2278],{},[87,2277,318],{"stretchy":89},[48,2279,2016],{},[87,2281,472],{},[1187,2283,2284,2286],{},[48,2285,2082],{},[48,2287,2016],{},[458,2289,654],{},[87,2291,2292],{},"×",[48,2294,2295],{},"0.729",[1053,2297,2298],{},[1056,2299,2300],{},[1059,2301,2302],{"scriptlevel":1061,"displaystyle":59},[45,2303,2304,2310,2312,2318,2320,2326,2328,2334,2336,2338,2344,2346,2352,2354,2356,2358,2360,2366,2368,2374,2376,2378],{},[1065,2305,2306,2308],{},[458,2307,2075],{},[462,2309,464],{},[87,2311,472],{},[1065,2313,2314,2316],{},[458,2315,2075],{},[462,2317,1118],{},[87,2319,93],{},[1065,2321,2322,2324],{},[458,2323,2075],{},[462,2325,1071],{},[87,2327,472],{},[1187,2329,2330,2332],{},[48,2331,2082],{},[48,2333,2016],{},[458,2335,654],{},[87,2337,1078],{},[656,2339,2340,2342],{},[48,2341,50],{},[48,2343,2016],{},[87,2345,93],{},[1187,2347,2348,2350],{},[48,2349,2082],{},[48,2351,2016],{},[458,2353,654],{},[87,2355,1078],{},[87,2357,370],{"stretchy":89},[48,2359,194],{},[656,2361,2362,2364],{},[87,2363,318],{"stretchy":89},[48,2365,2016],{},[87,2367,472],{},[1187,2369,2370,2372],{},[48,2371,2082],{},[48,2373,2016],{},[458,2375,654],{},[87,2377,2292],{},[48,2379,2380],{},"0.271",[1053,2382,2383],{},[1056,2384,2385],{},[1059,2386,2387],{"scriptlevel":1061,"displaystyle":59},[45,2388,2389,2403,2405,2419,2421,2424],{},[455,2390,2391,2393,2395],{},[458,2392,460],{},[462,2394,464],{},[45,2396,2397,2399,2401],{},[458,2398,469],{},[87,2400,472],{},[48,2402,2016],{},[87,2404,472],{},[1187,2406,2407,2413],{},[1065,2408,2409,2411],{},[458,2410,2075],{},[462,2412,464],{},[1065,2414,2415,2417],{},[458,2416,2075],{},[462,2418,1118],{},[87,2420,472],{},[48,2422,2423],{},"27.1",[458,2425,483],{"mathvariant":482},[52,2427,2428],{"encoding":54},"\\begin{aligned}\nV_\\text{inner} = \\frac{4}{3} \\pi \\cdot (0.9)^3 = \\frac{4}{3} \\pi \\times 0.729  \\\\\nV_\\text{shell} = V_\\text{unit} - V_\\text{inner} = \\frac{4}{3} \\pi \\cdot 1^3 - \\frac{4}{3} \\pi \\cdot (0.9)^{3} = \\frac{4}{3} \\pi \\times 0.271 \\\\\nP^{d=3}_\\text{shell} = \\frac{V_\\text{shell}}{V_\\text{unit}} = 27.1\\%  \n\\end{aligned}",[28,2430,2432],{"className":2431,"ariaHidden":59},[58],[28,2433,2435,2439],{"className":2434},[63],[28,2436],{"className":2437,"style":2438},[67],"height:7.1112em;vertical-align:-3.3056em;",[28,2440,2442],{"className":2441},[72],[28,2443,2445],{"className":2444},[1047],[28,2446,2448],{"className":2447},[1242],[28,2449,2451,3429],{"className":2450},[512,513],[28,2452,2454,3426],{"className":2453},[517],[28,2455,2458,2715,3190],{"className":2456,"style":2457},[521],"height:3.8056em;",[28,2459,2461,2464],{"style":2460},"top:-5.8445em;",[28,2462],{"className":2463,"style":1259},[529],[28,2465,2467,2511,2514,2517,2520,2582,2585,2588,2591,2594,2597,2600,2629,2632,2635,2638,2700,2703,2706,2709,2712],{"className":2466},[72],[28,2468,2470,2473],{"className":2469},[72],[28,2471,2075],{"className":2472,"style":2108},[72,503],[28,2474,2476],{"className":2475},[508],[28,2477,2479,2503],{"className":2478},[512,513],[28,2480,2482,2500],{"className":2481},[517],[28,2483,2485],{"className":2484,"style":1281},[521],[28,2486,2488,2491],{"style":2487},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[28,2489],{"className":2490,"style":530},[529],[28,2492,2494],{"className":2493},[534,535,536,537],[28,2495,2497],{"className":2496},[72,541,537],[28,2498,1071],{"className":2499},[72,537],[28,2501,571],{"className":2502},[570],[28,2504,2506],{"className":2505},[517],[28,2507,2509],{"className":2508,"style":1306},[521],[28,2510],{},[28,2512],{"className":2513,"style":584},[134],[28,2515,472],{"className":2516},[563],[28,2518],{"className":2519,"style":584},[134],[28,2521,2523,2526,2579],{"className":2522},[72],[28,2524],{"className":2525},[120,1753],[28,2527,2529],{"className":2528},[1187],[28,2530,2532,2571],{"className":2531},[512,513],[28,2533,2535,2568],{"className":2534},[517],[28,2536,2538,2549,2557],{"className":2537,"style":1925},[521],[28,2539,2540,2543],{"style":1769},[28,2541],{"className":2542,"style":1773},[529],[28,2544,2546],{"className":2545},[72],[28,2547,2016],{"className":2548},[72],[28,2550,2551,2554],{"style":1822},[28,2552],{"className":2553,"style":1773},[529],[28,2555],{"className":2556,"style":1830},[1829],[28,2558,2559,2562],{"style":1833},[28,2560],{"className":2561,"style":1773},[529],[28,2563,2565],{"className":2564},[72],[28,2566,2082],{"className":2567},[72],[28,2569,571],{"className":2570},[570],[28,2572,2574],{"className":2573},[517],[28,2575,2577],{"className":2576,"style":1968},[521],[28,2578],{},[28,2580],{"className":2581},[142,1753],[28,2583,654],{"className":2584,"style":699},[72,503],[28,2586],{"className":2587,"style":810},[134],[28,2589,1078],{"className":2590},[814],[28,2592],{"className":2593,"style":810},[134],[28,2595,370],{"className":2596},[120],[28,2598,194],{"className":2599},[72],[28,2601,2603,2606],{"className":2602},[142],[28,2604,318],{"className":2605},[142],[28,2607,2609],{"className":2608},[508],[28,2610,2612],{"className":2611},[512],[28,2613,2615],{"className":2614},[517],[28,2616,2618],{"className":2617,"style":1354},[521],[28,2619,2620,2623],{"style":1357},[28,2621],{"className":2622,"style":530},[529],[28,2624,2626],{"className":2625},[534,535,536,537],[28,2627,2016],{"className":2628},[72,537],[28,2630],{"className":2631,"style":584},[134],[28,2633,472],{"className":2634},[563],[28,2636],{"className":2637,"style":584},[134],[28,2639,2641,2644,2697],{"className":2640},[72],[28,2642],{"className":2643},[120,1753],[28,2645,2647],{"className":2646},[1187],[28,2648,2650,2689],{"className":2649},[512,513],[28,2651,2653,2686],{"className":2652},[517],[28,2654,2656,2667,2675],{"className":2655,"style":1925},[521],[28,2657,2658,2661],{"style":1769},[28,2659],{"className":2660,"style":1773},[529],[28,2662,2664],{"className":2663},[72],[28,2665,2016],{"className":2666},[72],[28,2668,2669,2672],{"style":1822},[28,2670],{"className":2671,"style":1773},[529],[28,2673],{"className":2674,"style":1830},[1829],[28,2676,2677,2680],{"style":1833},[28,2678],{"className":2679,"style":1773},[529],[28,2681,2683],{"className":2682},[72],[28,2684,2082],{"className":2685},[72],[28,2687,571],{"className":2688},[570],[28,2690,2692],{"className":2691},[517],[28,2693,2695],{"className":2694,"style":1968},[521],[28,2696],{},[28,2698],{"className":2699},[142,1753],[28,2701,654],{"className":2702,"style":699},[72,503],[28,2704],{"className":2705,"style":810},[134],[28,2707,2292],{"className":2708},[814],[28,2710],{"className":2711,"style":810},[134],[28,2713,2295],{"className":2714},[72],[28,2716,2718,2721],{"style":2717},"top:-3.5371em;",[28,2719],{"className":2720,"style":1259},[529],[28,2722,2724,2767,2770,2773,2776,2819,2822,2825,2828,2871,2874,2877,2880,2942,2945,2948,2951,2954,2983,2986,2989,2992,3054,3057,3060,3063,3066,3069,3072,3104,3107,3110,3113,3175,3178,3181,3184,3187],{"className":2723},[72],[28,2725,2727,2730],{"className":2726},[72],[28,2728,2075],{"className":2729,"style":2108},[72,503],[28,2731,2733],{"className":2732},[508],[28,2734,2736,2759],{"className":27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ssName":3068},[120],[28,3070,194],{"className":3071},[72],[28,3073,3075,3078],{"className":3074},[142],[28,3076,318],{"className":3077},[142],[28,3079,3081],{"className":3080},[508],[28,3082,3084],{"className":3083},[512],[28,3085,3087],{"className":3086},[517],[28,3088,3090],{"className":3089,"style":1354},[521],[28,3091,3092,3095],{"style":1357},[28,3093],{"className":3094,"style":530},[529],[28,3096,3098],{"className":3097},[534,535,536,537],[28,3099,3101],{"className":3100},[72,537],[28,3102,2016],{"className":3103},[72,537],[28,3105],{"className":3106,"style":584},[134],[28,3108,472],{"className":3109},[563],[28,3111],{"className":3112,"style":584},[134],[28,3114,3116,3119,3172],{"className":3115},[72],[28,3117],{"className":3118},[120,1753],[28,3120,3122],{"className":3121},[1187],[28,3123,3125,3164],{"className":3124},[512,513],[28,3126,3128,3161],{"className":3127},[517],[28,3129,3131,3142,3150],{"className":3130,"style":1925},[521],[28,3132,3133,3136],{"style":1769},[28,3134],{"className":3135,"style":1773},[529],[28,3137,3139],{"className":3138},[72],[28,3140,2016],{"className":3141},[72],[28,3143,3144,3147],{"style":1822},[28,3145],{"className":3146,"style":1773},[529],[28,3148],{"className":3149,"style":1830},[1829],[28,3151,3152,3155],{"style":1833},[28,3153],{"className":3154,"style":1773},[529],[28,3156,3158],{"className":3157},[72],[28,3159,2082],{"className":3160},[72],[28,3162,571],{"className":3163},[570],[28,3165,3167],{"className":3166},[517],[28,3168,3170],{"className":3169,"style":1968},[521],[28,3171],{},[28,3173],{"className":3174},[142,1753],[28,3176,654],{"className":3177,"style":699},[72,503],[28,3179],{"className":3180,"style":810},[134],[28,3182,2292],{"className":3183},[814],[28,3185],{"className":3186,"style":810},[134],[28,3188,2380],{"className":3189},[72],[28,3191,3193,3196],{"style":3192},"top:-1.1907em;",[28,3194],{"className":3195,"style":1259},[529],[28,3197,3199,3262,3265,3268,3271,3413,3416,3419,3422],{"className":3198},[72],[28,3200,3202,3205],{"className":3201},[72],[28,3203,460],{"className":3204,"style":504},[72,503],[28,3206,3208],{"className":3207},[508],[28,3209,3211,3254],{"className":3210},[512,513],[28,3212,3214,3251],{"className":3213},[517],[28,3215,3217,3231],{"className":3216,"style":1690},[521],[28,3218,3219,3222],{"style":1693},[28,3220],{"className":3221,"style":530},[529],[28,3223,3225],{"className":3224},[534,535,536,537],[28,3226,3228],{"className":3227},[72,541,537],[28,3229,464],{"className":3230},[72,537],[28,3232,3233,3236],{"style":1357},[28,3234],{"className":3235,"style":530},[529],[28,3237,3239],{"className":3238},[534,535,536,537],[28,3240,3242,3245,3248],{"className":3241},[72,537],[28,3243,469],{"className":3244},[72,503,537],[28,3246,472],{"className":3247},[563,537],[28,3249,2016],{"className":3250},[72,537],[28,3252,571],{"className":3253},[570],[28,3255,3257],{"className":3256},[517],[28,3258,3260],{"className":3259,"style":1735},[521],[28,3261],{},[28,3263],{"className":3264,"style":584},[134],[28,3266,472],{"className":3267},[563],[28,3269],{"className":3270,"style":584},[134],[28,3272,3274,3277,3410],{"className":3273},[72],[28,3275],{"className":3276},[120,1753],[28,3278,3280],{"className":3279},[1187],[28,3281,3283,3402],{"className":3282},[512,513],[28,3284,3286,3399],{"className":3285},[517],[28,3287,3289,3340,3348],{"className":3288,"style":1766},[521],[28,3290,3291,3294],{"style":1769},[28,3292],{"className":3293,"style":1773},[529],[28,3295,3297],{"className":3296},[72],[28,3298,3300,3303],{"className":3299},[72],[28,3301,2075],{"className":3302,"style":2108},[72,503],[28,3304,3306],{"className":3305},[508],[28,3307,3309,3332],{"className":3308},[512,513],[28,3310,3312,3329],{"className":3311},[517],[28,3313,3315],{"className":3314,"style":1281},[521],[28,3316,3317,3320],{"style":2487},[28,3318],{"className":3319,"style":530},[529],[28,3321,3323],{"className":3322},[534,535,536,537],[28,3324,3326],{"className":3325},[72,541,537],[28,3327,1118],{"className":3328},[72,537],[28,3330,571],{"className":3331},[570],[28,3333,3335],{"className":3334},[517],[28,3336,3338],{"className":3337,"style":1306},[521],[28,3339],{},[28,3341,3342,3345],{"style":1822},[28,3343],{"className":3344,"style":1773},[529],[28,3346],{"className":3347,"style":1830},[1829],[28,3349,3350,3353],{"style":1833},[28,3351],{"className":3352,"style":1773},[529],[28,3354,3356],{"className":3355},[72],[28,3357,3359,3362],{"className":3358},[72],[28,3360,2075],{"className":3361,"style":2108},[72,503],[28,3363,3365],{"className":3364},[508],[28,3366,3368,3391],{"className":3367},[512,513],[28,3369,3371,3388],{"className":3370},[517],[28,3372,3374],{"className":3373,"style":1409},[521],[28,3375,3376,3379],{"style":2487},[28,3377],{"className":3378,"style":530},[529],[28,3380,3382],{"className":3381},[534,535,536,537],[28,3383,3385],{"className":3384},[72,541,537],[28,3386,464],{"className":3387},[72,537],[28,3389,571],{"className":3390},[570],[28,3392,3394],{"className":3393},[517],[28,3395,3397],{"className":3396,"style":1306},[521],[28,3398],{},[28,3400,571],{"className":3401},[570],[28,3403,3405],{"className":3404},[517],[28,3406,3408],{"className":3407,"style":1892},[521],[28,3409],{},[28,3411],{"className":3412},[142,1753],[28,3414],{"className":3415,"style":584},[134],[28,3417,472],{"className":3418},[563],[28,3420],{"className":3421,"style":584},[134],[28,3423,3425],{"className":3424},[72],"27.1%",[28,3427,571],{"className":3428},[570],[28,3430,3432],{"className":3431},[517],[28,3433,3436],{"className":3434,"style":3435},[521],"height:3.3056em;",[28,3437],{},[10,3439,3440,3441,3497,3498,409],{},"Here, we seem to be able to summarize a general pattern: as the dimension increases, the proportion of samples distributed on the outer shell of the space increases continuously ",[28,3442,3444,3467],{"className":3443,"translate":32},[31],[28,3445,3447],{"className":3446},[36],[38,3448,3449],{"xmlns":40},[42,3450,3451,3464],{},[45,3452,3453,3455,3457,3460,3462],{},[48,3454,479],{},[458,3456,483],{"mathvariant":482},[87,3458,3459],{},"→",[48,3461,2423],{},[458,3463,483],{"mathvariant":482},[52,3465,3466],{"encoding":54},"10\\% \\rightarrow 27.1\\%",[28,3468,3470,3488],{"className":3469,"ariaHidden":59},[58],[28,3471,3473,3476,3479,3482,3485],{"className":3472},[63],[28,3474],{"className":3475,"style":597},[67],[28,3477,601],{"className":3478},[72],[28,3480],{"className":3481,"style":584},[134],[28,3483,3459],{"className":3484},[563],[28,3486],{"className":3487,"style":584},[134],[28,3489,3491,3494],{"className":3490},[63],[28,3492],{"className":3493,"style":597},[67],[28,3495,3425],{"className":3496},[72],", while the proportion of samples in the cavity decreases continuously ",[28,3499,3501,3525],{"className":3500,"translate":32},[31],[28,3502,3504],{"className":3503},[36],[38,3505,3506],{"xmlns":40},[42,3507,3508,3522],{},[45,3509,3510,3513,3515,3517,3520],{},[48,3511,3512],{},"90",[458,3514,483],{"mathvariant":482},[87,3516,3459],{},[48,3518,3519],{},"72.9",[458,3521,483],{"mathvariant":482},[52,3523,3524],{"encoding":54},"90\\% \\rightarrow 72.9\\%",[28,3526,3528,3547],{"className":3527,"ariaHidden":59},[58],[28,3529,3531,3534,3538,3541,3544],{"className":3530},[63],[28,3532],{"className":3533,"style":597},[67],[28,3535,3537],{"className":3536},[72],"90%",[28,3539],{"className":3540,"style":584},[134],[28,3542,3459],{"className":3543},[563],[28,3545],{"className":3546,"style":584},[134],[28,3548,3550,3553],{"className":3549},[63],[28,3551],{"className":3552,"style":597},[67],[28,3554,3556],{"className":3555},[72],"72.9%",[10,3558,3559,3560,3589,3590,3619],{},"To rigorously verify this conclusion, we need general mathematical reasoning. For a ",[28,3561,3563,3576],{"className":3562,"translate":32},[31],[28,3564,3566],{"className":3565},[36],[38,3567,3568],{"xmlns":40},[42,3569,3570,3574],{},[45,3571,3572],{},[458,3573,469],{},[52,3575,469],{"encoding":54},[28,3577,3579],{"className":3578,"ariaHidden":59},[58],[28,3580,3582,3586],{"className":3581},[63],[28,3583],{"className":3584,"style":3585},[67],"height:0.6944em;",[28,3587,469],{"className":3588},[72,503],"-dimensional space, the volume of a hypersphere with radius ",[28,3591,3593,3606],{"className":3592,"translate":32},[31],[28,3594,3596],{"className":3595},[36],[38,3597,3598],{"xmlns":40},[42,3599,3600,3604],{},[45,3601,3602],{},[458,3603,660],{},[52,3605,660],{"encoding":54},[28,3607,3609],{"className":3608,"ariaHidden":59},[58],[28,3610,3612,3616],{"className":3611},[63],[28,3613],{"className":3614,"style":3615},[67],"height:0.4306em;",[28,3617,660],{"className":3618,"style":706},[72,503]," is:",[28,3621,3623],{"className":3622,"translate":32},[1034],[28,3624,3626,3694],{"className":3625,"translate":32},[31],[28,3627,3629],{"className":3628},[36],[38,3630,3631],{"xmlns":40,"display":1043},[42,3632,3633,3691],{},[45,3634,3635,3641,3643,3645,3647,3649,3685],{},[656,3636,3637,3639],{},[458,3638,2075],{},[458,3640,469],{},[87,3642,370],{"stretchy":89},[458,3644,660],{},[87,3646,318],{"stretchy":89},[87,3648,472],{},[1187,3650,3651,3664],{},[656,3652,3653,3655],{},[458,3654,654],{},[45,3656,3657,3659,3662],{},[458,3658,469],{},[458,3660,3661],{"mathvariant":482},"\u002F",[48,3663,619],{},[45,3665,3666,3669],{},[458,3667,3668],{"mathvariant":482},"Γ",[45,3670,3671,3673,3679,3681,3683],{},[87,3672,370],{"fence":59},[1187,3674,3675,3677],{},[458,3676,469],{},[48,3678,619],{},[87,3680,752],{},[48,3682,50],{},[87,3684,318],{"fence":59},[656,3686,3687,3689],{},[458,3688,660],{},[458,3690,469],{},[52,3692,3693],{"encoding":54}," V^d(r) = 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r^d",[28,3695,3697,3751],{"className":3696,"ariaHidden":59},[58],[28,3698,3700,3704,3733,3736,3739,3742,3745,3748],{"className":3699},[63],[28,3701],{"className":3702,"style":3703},[67],"height:1.1491em;vertical-align:-0.25em;",[28,3705,3707,3710],{"className":3706},[72],[28,3708,2075],{"className":3709,"style":2108},[72,503],[28,3711,3713],{"className":3712},[508],[28,3714,3716],{"className":3715},[512],[28,3717,3719],{"className":3718},[517],[28,3720,3722],{"className":3721,"style":1690},[521],[28,3723,3724,3727],{"style":1357},[28,3725],{"className":3726,"style":530},[529],[28,3728,3730],{"className":3729},[534,535,536,537],[28,3731,469],{"className":3732},[72,503,537],[28,3734,370],{"className":3735},[120],[28,3737,660],{"className":3738,"style":706},[72,503],[28,3740,318],{"className":3741},[142],[28,3743],{"className":3744,"style":584},[134],[28,3746,472],{"className":3747},[563],[28,3749],{"className":3750,"style":584},[134],[28,3752,3754,3758,3961],{"className":3753},[63],[28,3755],{"className":3756,"style":3757},[67],"height:2.6851em;vertical-align:-1.1201em;",[28,3759,3761,3764,3958],{"className":3760},[72],[28,3762],{"className":3763},[120,1753],[28,3765,3767],{"className":3766},[1187],[28,3768,3770,3949],{"className":3769},[512,513],[28,3771,3773,3946],{"className":3772},[517],[28,3774,3777,3893,3901],{"className":3775,"style":3776},[521],"height:1.565em;",[28,3778,3780,3783],{"style":3779},"top:-2.2299em;",[28,3781],{"className":3782,"style":1773},[529],[28,3784,3786,3789,3792],{"className":3785},[72],[28,3787,3668],{"className":3788},[72],[28,3790],{"className":3791,"style":135},[134],[28,3793,3796,3806,3875,3878,3881,3884,3887],{"className":3794},[3795],"minner",[28,3797,3801],{"className":3798,"style":3800},[120,3799],"delimcenter","top:0em;",[28,3802,370],{"className":3803},[3804,3805],"delimsizing","size1",[28,3807,3809,3812,3872],{"className":3808},[72],[28,3810],{"className":3811},[120,1753],[28,3813,3815],{"className":3814},[1187],[28,3816,3818,3864],{"className":3817},[512,513],[28,3819,3821,3861],{"className":3820},[517],[28,3822,3825,3839,3847],{"className":3823,"style":3824},[521],"height:0.8801em;",[28,3826,3827,3830],{"style":2146},[28,3828],{"className":3829,"style":1773},[529],[28,3831,3833],{"className":3832},[534,535,536,537],[28,3834,3836],{"className":3835},[72,537],[28,3837,619],{"className":3838},[72,537],[28,3840,3841,3844],{"style":1822},[28,3842],{"className":3843,"style":1773},[529],[28,3845],{"className":3846,"style":1830},[1829],[28,3848,3849,3852],{"style":2169},[28,3850],{"className":3851,"style":1773},[529],[28,3853,3855],{"className":3854},[534,535,536,537],[28,3856,3858],{"className":3857},[72,537],[28,3859,469],{"className":3860},[72,503,537],[28,3862,571],{"className":3863},[570],[28,3865,3867],{"className":3866},[517],[28,3868,3870],{"className":3869,"style":2191},[521],[28,3871],{},[28,3873],{"className":3874},[142,1753],[28,3876],{"className":3877,"style":810},[134],[28,3879,752],{"className":3880},[814],[28,3882],{"className":3883,"style":810},[134],[28,3885,50],{"className":3886},[72],[28,3888,3890],{"className":3889,"style":3800},[142,3799],[28,3891,318],{"className":3892},[3804,3805],[28,3894,3895,3898],{"style":1822},[28,3896],{"className":3897,"style":1773},[529],[28,3899],{"className":3900,"style":1830},[1829],[28,3902,3903,3906],{"style":1833},[28,3904],{"className":3905,"style":1773},[529],[28,3907,3909],{"className":3908},[72],[28,3910,3912,3915],{"className":3911},[72],[28,3913,654],{"className":3914,"style":699},[72,503],[28,3916,3918],{"className":3917},[508],[28,3919,3921],{"className":3920},[512],[28,3922,3924],{"className":3923},[517],[28,3925,3928],{"className":3926,"style":3927},[521],"height:0.888em;",[28,3929,3930,3933],{"style":547},[28,3931],{"className":3932,"style":530},[529],[28,3934,3936],{"className":3935},[534,535,536,537],[28,3937,3939,3942],{"className":3938},[72,537],[28,3940,469],{"className":3941},[72,503,537],[28,3943,3945],{"className":3944},[72,537],"\u002F2",[28,3947,571],{"className":3948},[570],[28,3950,3952],{"className":3951},[517],[28,3953,3956],{"className":3954,"style":3955},[521],"height:1.1201em;",[28,3957],{},[28,3959],{"className":3960},[142,1753],[28,3962,3964,3967],{"className":3963},[72],[28,3965,660],{"className":3966,"style":706},[72,503],[28,3968,3970],{"className":3969},[508],[28,3971,3973],{"className":3972},[512],[28,3974,3976],{"className":3975},[517],[28,3977,3979],{"className":3978,"style":1690},[521],[28,3980,3981,3984],{"style":1357},[28,3982],{"className":3983,"style":530},[529],[28,3985,3987],{"className":3986},[534,535,536,537],[28,3988,469],{"className":3989},[72,503,537],[10,3991,3992],{},"Where:",[3994,3995,3996,4044,4368],"ul",{},[3997,3998,3999,4043],"li",{},[28,4000,4002,4022],{"className":4001,"translate":32},[31],[28,4003,4005],{"className":4004},[36],[38,4006,4007],{"xmlns":40},[42,4008,4009,4019],{},[45,4010,4011,4013,4015,4017],{},[458,4012,3668],{"mathvariant":482},[87,4014,370],{"stretchy":89},[87,4016,1078],{},[87,4018,318],{"stretchy":89},[52,4020,4021],{"encoding":54},"\\Gamma(\\cdot)",[28,4023,4025],{"className":4024,"ariaHidden":59},[58],[28,4026,4028,4031,4034,4037,4040],{"className":4027},[63],[28,4029],{"className":4030,"style":116},[67],[28,4032,3668],{"className":4033},[72],[28,4035,370],{"className":4036},[120],[28,4038,1078],{"className":4039},[72],[28,4041,318],{"className":4042},[142]," is the gamma function;",[3997,4045,4046,4047,4075,4076,4338,4339,4367],{},"When ",[28,4048,4050,4063],{"className":4049,"translate":32},[31],[28,4051,4053],{"className":4052},[36],[38,4054,4055],{"xmlns":40},[42,4056,4057,4061],{},[45,4058,4059],{},[458,4060,469],{},[52,4062,469],{"encoding":54},[28,4064,4066],{"className":4065,"ariaHidden":59},[58],[28,4067,4069,4072],{"className":4068},[63],[28,4070],{"className":4071,"style":3585},[67],[28,4073,469],{"className":4074},[72,503]," is a positive integer, ",[28,4077,4079,4126],{"className":4078,"translate":32},[31],[28,4080,4082],{"className":4081},[36],[38,4083,4084],{"xmlns":40},[42,4085,4086,4123],{},[45,4087,4088,4090,4106,4108,4120],{},[458,4089,3668],{"mathvariant":482},[45,4091,4092,4094,4100,4102,4104],{},[87,4093,370],{"fence":59},[1187,4095,4096,4098],{},[458,4097,469],{},[48,4099,619],{},[87,4101,752],{},[48,4103,50],{},[87,4105,318],{"fence":59},[87,4107,472],{},[45,4109,4110,4112,4118],{},[87,4111,370],{"fence":59},[1187,4113,4114,4116],{},[458,4115,469],{},[48,4117,619],{},[87,4119,318],{"fence":59},[87,4121,4122],{"stretchy":89},"!",[52,4124,4125],{"encoding":54},"\\Gamma\\left(\\frac{d}{2}+1\\right) = \\left(\\frac{d}{2}\\right)!",[28,4127,4129,4246],{"className":4128,"ariaHidden":59},[58],[28,4130,4132,4136,4139,4142,4237,4240,4243],{"className":4131},[63],[28,4133],{"className":4134,"style":4135},[67],"height:1.2301em;vertical-align:-0.35em;",[28,4137,3668],{"className":4138},[72],[28,4140],{"className":4141,"style":135},[134],[28,4143,4145,4151,4219,4222,4225,4228,4231],{"className":4144},[3795],[28,4146,4148],{"className":4147,"style":3800},[120,3799],[28,4149,370],{"className":4150},[3804,3805],[28,4152,4154,4157,4216],{"className":4153},[72],[28,4155],{"className":4156},[120,1753],[28,4158,4160],{"className":4159},[1187],[28,4161,4163,4208],{"className":4162},[512,513],[28,4164,4166,4205],{"className":4165},[517],[28,4167,4169,4183,4191],{"className":4168,"style":3824},[521],[28,4170,4171,4174],{"style":2146},[28,4172],{"className":4173,"style":1773},[529],[28,4175,4177],{"className":4176},[534,535,536,537],[28,4178,4180],{"className":4179},[72,537],[28,4181,619],{"className":4182},[72,537],[28,4184,4185,4188],{"style":1822},[28,4186],{"className":4187,"style":1773},[529],[28,4189],{"className":4190,"style":1830},[1829],[28,4192,4193,4196],{"style":2169},[28,4194],{"className":4195,"style":1773},[529],[28,4197,4199],{"className":4198},[534,535,536,537],[28,4200,4202],{"className":4201},[72,537],[28,4203,469],{"className":4204},[72,503,537],[28,4206,571],{"className":4207},[570],[28,4209,4211],{"className":4210},[517],[28,4212,4214],{"className":4213,"style":2191},[521],[28,4215],{},[28,4217],{"className":4218},[142,1753],[28,4220],{"className":4221,"style":810},[134],[28,4223,752],{"className":4224},[814],[28,4226],{"className":4227,"style":810},[134],[28,4229,50],{"className":4230},[72],[28,4232,4234],{"className":4233,"style":3800},[142,3799],[28,4235,318],{"className":4236},[3804,3805],[28,4238],{"className":4239,"style":584},[134],[28,4241,472],{"className":4242},[563],[28,4244],{"className":4245,"style":584},[134],[28,4247,4249,4252,4335],{"className":4248},[63],[28,4250],{"className":4251,"style":4135},[67],[28,4253,4255,4261,4329],{"className":4254},[3795],[28,4256,4258],{"className":4257,"style":3800},[120,3799],[28,4259,370],{"className":4260},[3804,3805],[28,4262,4264,4267,4326],{"className":4263},[72],[28,4265],{"className":4266},[120,1753],[28,4268,4270],{"className":4269},[1187],[28,4271,4273,4318],{"className":4272},[512,513],[28,4274,4276,4315],{"className":4275},[517],[28,4277,4279,4293,4301],{"className":4278,"style":3824},[521],[28,4280,4281,4284],{"style":2146},[28,4282],{"className":4283,"style":1773},[529],[28,4285,4287],{"className":4286},[534,535,536,537],[28,4288,4290],{"className":4289},[72,537],[28,4291,619],{"className":4292},[72,537],[28,4294,4295,4298],{"style":1822},[28,4296],{"className":4297,"style":1773},[529],[28,4299],{"className":4300,"style":1830},[1829],[28,4302,4303,4306],{"style":2169},[28,4304],{"className":4305,"style":1773},[529],[28,4307,4309],{"className":4308},[534,535,536,537],[28,4310,4312],{"className":4311},[72,537],[28,4313,469],{"className":4314},[72,503,537],[28,4316,571],{"className":4317},[570],[28,4319,4321],{"className":4320},[517],[28,4322,4324],{"className":4323,"style":2191},[521],[28,4325],{},[28,4327],{"className":4328},[142,1753],[28,4330,4332],{"className":4331,"style":3800},[142,3799],[28,4333,318],{"className":4334},[3804,3805],[28,4336,4122],{"className":4337},[142],", 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",[28,4401,4403,4420],{"className":4402,"translate":32},[31],[28,4404,4406],{"className":4405},[36],[38,4407,4408],{"xmlns":40},[42,4409,4410,4417],{},[45,4411,4412],{},[4413,4414,4415],"msqrt",{},[458,4416,654],{},[52,4418,4419],{"encoding":54},"\\sqrt{\\pi}",[28,4421,4423],{"className":4422,"ariaHidden":59},[58],[28,4424,4426,4430],{"className":4425},[63],[28,4427],{"className":4428,"style":4429},[67],"height:1.04em;vertical-align:-0.2397em;",[28,4431,4434],{"className":4432},[72,4433],"sqrt",[28,4435,4437,4485],{"className":4436},[512,513],[28,4438,4440,4482],{"className":4439},[517],[28,4441,4444,4459],{"className":4442,"style":4443},[521],"height:0.8003em;",[28,4445,4449,4452],{"className":4446,"style":4448},[4447],"svg-align","top:-3em;",[28,4450],{"className":4451,"style":1773},[529],[28,4453,4456],{"className":4454,"style":4455},[72],"padding-left:0.833em;",[28,4457,654],{"className":4458,"style":699},[72,503],[28,4460,4462,4465],{"style":4461},"top:-2.7603em;",[28,4463],{"className":4464,"style":1773},[529],[28,4466,4470],{"className":4467,"style":4469},[4468],"hide-tail","min-width:0.853em;height:1.08em;",[4471,4472,4478],"svg",{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[4479,4480],"path",{"d":4481},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[28,4483,571],{"className":4484},[570],[28,4486,4488],{"className":4487},[517],[28,4489,4492],{"className":4490,"style":4491},[521],"height:0.2397em;",[28,4493],{},[10,4495,4496],{},"We have:",[28,4498,4500],{"className":4499,"translate":32},[1034],[28,4501,4503,4563],{"className":4502,"translate":32},[31],[28,4504,4506],{"className":4505},[36],[38,4507,4508],{"xmlns":40,"display":1043},[42,4509,4510,4560],{},[45,4511,4512,4520,4522,4540,4542,4548,4550,4552,4554],{},[455,4513,4514,4516,4518],{},[458,4515,460],{},[462,4517,464],{},[458,4519,469],{},[87,4521,472],{},[1187,4523,4524,4532],{},[455,4525,4526,4528,4530],{},[458,4527,2075],{},[462,4529,464],{},[458,4531,469],{},[455,4533,4534,4536,4538],{},[458,4535,2075],{},[462,4537,1118],{},[458,4539,469],{},[87,4541,472],{},[656,4543,4544,4546],{},[48,4545,50],{},[458,4547,469],{},[87,4549,93],{},[87,4551,370],{"stretchy":89},[48,4553,194],{},[656,4555,4556,4558],{},[87,4557,318],{"stretchy":89},[458,4559,469],{},[52,4561,4562],{"encoding":54},"P^{d}_\\text{shell} = \\frac{V^d_\\text{shell}}{V^d_\\text{unit}} = \n1^d - (0.9)^d",[28,4564,4566,4639,4827,4872],{"className":4565,"ariaHidden":59},[58],[28,4567,4569,4573,4630,4633,4636],{"className":4568},[63],[28,4570],{"className":4571,"style":4572},[67],"height:1.1461em;vertical-align:-0.247em;",[28,4574,4576,4579],{"className":4575},[72],[28,4577,460],{"className":4578,"style":504},[72,503],[28,4580,4582],{"className":4581},[508],[28,4583,4585,4622],{"className":4584},[512,513],[28,4586,4588,4619],{"className":4587},[517],[28,4589,4591,4605],{"className":4590,"style":1690},[521],[28,4592,4593,4596],{"style":1693},[28,4594],{"className":4595,"style":530},[529],[28,4597,4599],{"className":4598},[534,535,536,537],[28,4600,4602],{"className":4601},[72,541,537],[28,4603,464],{"className":4604},[72,537],[28,4606,4607,4610],{"style":1357},[28,4608],{"className":4609,"style":530},[529],[28,4611,4613],{"className":4612},[534,535,536,537],[28,4614,4616],{"className":4615},[72,537],[28,4617,469],{"className":4618},[72,503,537],[28,4620,571],{"className":4621},[570],[28,4623,4625],{"className":4624},[517],[28,4626,4628],{"className":4627,"style":1735},[521],[28,4629],{},[28,4631],{"className":4632,"style":584},[134],[28,4634,472],{"className":4635},[563],[28,4637],{"className":4638,"style":584},[134],[28,4640,4642,4646,4818,4821,4824],{"className":4641},[63],[28,4643],{"className":4644,"style":4645},[67],"height:2.5297em;vertical-align:-1.0036em;",[28,4647,4649,4652,4815],{"className":4648},[72],[28,4650],{"className":4651},[120,1753],[28,4653,4655],{"className":4654},[1187],[28,4656,4658,4806],{"className":4657},[512,513],[28,4659,4661,4803],{"className":4660},[517],[28,4662,4665,4732,4740],{"className":4663,"style":4664},[521],"height:1.5261em;",[28,4666,4668,4671],{"style":4667},"top:-2.2791em;",[28,4669],{"className":4670,"style":1773},[529],[28,4672,4674],{"className":4673},[72],[28,4675,4677,4680],{"className":4676},[72],[28,4678,2075],{"className":4679,"style":2108},[72,503],[28,4681,4683],{"className":4682},[508],[28,4684,4686,4723],{"className":4685},[512,513],[28,4687,4689,4720],{"className":4688},[517],[28,4690,4693,4708],{"className":4691,"style":4692},[521],"height:0.8309em;",[28,4694,4696,4699],{"style":4695},"top:-2.4173em;margin-left:-0.2222em;margin-right:0.05em;",[28,4697],{"className":4698,"style":530},[529],[28,4700,4702],{"className":4701},[534,535,536,537],[28,4703,4705],{"className":4704},[72,541,537],[28,4706,1118],{"className":4707},[72,537],[28,4709,4711,4714],{"style":4710},"top:-3.0448em;margin-right:0.05em;",[28,4712],{"className":4713,"style":530},[529],[28,4715,4717],{"className":4716},[534,535,536,537],[28,4718,469],{"className":4719},[72,503,537],[28,4721,571],{"className":4722},[570],[28,4724,4726],{"className":4725},[517],[28,4727,4730],{"className":4728,"style":4729},[521],"height:0.2827em;",[28,4731],{},[28,4733,4734,4737],{"style":1822},[28,4735],{"className":4736,"style":1773},[529],[28,4738],{"className":4739,"style":1830},[1829],[28,4741,4742,4745],{"style":1833},[28,4743],{"className":4744,"style":1773},[529],[28,4746,4748],{"className":4747},[72],[28,4749,4751,4754],{"className":4750},[72],[28,4752,2075],{"className":4753,"style":2108},[72,503],[28,4755,4757],{"className":4756},[508],[28,4758,4760,4795],{"className":4759},[512,513],[28,4761,4763,4792],{"className":4762},[517],[28,4764,4766,4781],{"className":4765,"style":522},[521],[28,4767,4769,4772],{"style":4768},"top:-2.4169em;margin-left:-0.2222em;margin-right:0.05em;",[28,4770],{"className":4771,"style":530},[529],[28,4773,4775],{"className":4774},[534,535,536,537],[28,4776,4778],{"className":4777},[72,541,537],[28,4779,464],{"className":4780},[72,537],[28,4782,4783,4786],{"style":547},[28,4784],{"className":4785,"style":530},[529],[28,4787,4789],{"className":4788},[534,535,536,537],[28,4790,469],{"className":4791},[72,503,537],[28,4793,571],{"className":4794},[570],[28,4796,4798],{"className":4797},[517],[28,4799,4801],{"className":4800,"style":578},[521],[28,4802],{},[28,4804,571],{"className":4805},[570],[28,4807,4809],{"className":4808},[517],[28,4810,4813],{"className":4811,"style":4812},[521],"height:1.0036em;",[28,4814],{},[28,4816],{"className":4817},[142,1753],[28,4819],{"className":4820,"style":584},[134],[28,4822,472],{"className":4823},[563],[28,4825],{"className":4826,"style":584},[134],[28,4828,4830,4834,4863,4866,4869],{"className":4829},[63],[28,4831],{"className":4832,"style":4833},[67],"height:0.9824em;vertical-align:-0.0833em;",[28,4835,4837,4840],{"className":4836},[72],[28,4838,50],{"className":4839},[72],[28,4841,4843],{"className":4842},[508],[28,4844,4846],{"className":4845},[512],[28,4847,4849],{"className":4848},[517],[28,4850,4852],{"className":4851,"style":1690},[521],[28,4853,4854,4857],{"style":1357},[28,4855],{"className":4856,"style":530},[529],[28,4858,4860],{"className":4859},[534,535,536,537],[28,4861,469],{"className":4862},[72,503,537],[28,4864],{"className":4865,"style":810},[134],[28,4867,93],{"className":4868},[814],[28,4870],{"className":4871,"style":810},[134],[28,4873,4875,4878,4881,4884],{"className":4874},[63],[28,4876],{"className":4877,"style":3703},[67],[28,4879,370],{"className":4880},[120],[28,4882,194],{"className":4883},[72],[28,4885,4887,4890],{"className":4886},[142],[28,4888,318],{"className":4889},[142],[28,4891,4893],{"className":4892},[508],[28,4894,4896],{"className":4895},[512],[28,4897,4899],{"className":4898},[517],[28,4900,4902],{"className":4901,"style":1690},[521],[28,4903,4904,4907],{"style":1357},[28,4905],{"className":4906,"style":530},[529],[28,4908,4910],{"className":4909},[534,535,536,537],[28,4911,469],{"className":4912},[72,503,537],[10,4914,4915,4916,4974],{},"For high dimensions ",[28,4917,4919,4940],{"className":4918,"translate":32},[31],[28,4920,4922],{"className":4921},[36],[38,4923,4924],{"xmlns":40},[42,4925,4926,4937],{},[45,4927,4928,4930,4932,4934],{},[458,4929,469],{},[87,4931,3459],{},[87,4933,752],{},[458,4935,4936],{"mathvariant":482},"∞",[52,4938,4939],{"encoding":54},"d \\rightarrow +\\infty",[28,4941,4943,4961],{"className":4942,"ariaHidden":59},[58],[28,4944,4946,4949,4952,4955,4958],{"className":4945},[63],[28,4947],{"className":4948,"style":3585},[67],[28,4950,469],{"className":4951},[72,503],[28,4953],{"className":4954,"style":584},[134],[28,4956,3459],{"className":4957},[563],[28,4959],{"className":4960,"style":584},[134],[28,4962,4964,4968,4971],{"className":4963},[63],[28,4965],{"className":4966,"style":4967},[67],"height:0.6667em;vertical-align:-0.0833em;",[28,4969,752],{"className":4970},[72],[28,4972,4936],{"className":4973},[72],", we have the limit:",[28,4976,4978],{"className":4977,"translate":32},[1034],[28,4979,4981,5044],{"className":4980,"translate":32},[31],[28,4982,4984],{"className":4983},[36],[38,4985,4986],{"xmlns":40,"display":1043},[42,4987,4988,5041],{},[45,4989,4990,5011,5017,5019,5021,5023,5029,5031,5033,5035,5037,5039],{},[4991,4992,4993,5001],"munder",{},[45,4994,4995,4998],{},[458,4996,4997],{},"lim",[87,4999,5000],{},"⁡",[45,5002,5003,5005,5007,5009],{},[458,5004,469],{},[87,5006,3459],{},[87,5008,752],{},[458,5010,4936],{"mathvariant":482},[656,5012,5013,5015],{},[48,5014,50],{},[458,5016,469],{},[87,5018,93],{},[87,5020,370],{"stretchy":89},[48,5022,194],{},[656,5024,5025,5027],{},[87,5026,318],{"stretchy":89},[458,5028,469],{},[87,5030,472],{},[48,5032,50],{},[87,5034,93],{},[48,5036,1061],{},[87,5038,472],{},[48,5040,50],{},[52,5042,5043],{"encoding":54},"\\lim_{d \\rightarrow +\\infty} 1^d - (0.9)^d = 1 - 0 = 1",[28,5045,5047,5155,5205,5224,5242],{"className":5046,"ariaHidden":59},[58],[28,5048,5050,5054,5114,5117,5146,5149,5152],{"className":5049},[63],[28,5051],{"className":5052,"style":5053},[67],"height:1.7095em;vertical-align:-0.8104em;",[28,5055,5059],{"className":5056},[5057,5058],"mop","op-limits",[28,5060,5062,5105],{"className":5061},[512,513],[28,5063,5065,5102],{"className":5064},[517],[28,5066,5068,5092],{"className":5067,"style":3585},[521],[28,5069,5071,5074],{"style":5070},"top:-2.3479em;margin-left:0em;",[28,5072],{"className":5073,"style":1773},[529],[28,5075,5077],{"className":5076},[534,535,536,537],[28,5078,5080,5083,5086,5089],{"className":5079},[72,537],[28,5081,469],{"className":5082},[72,503,537],[28,5084,3459],{"className":5085},[563,537],[28,5087,752],{"className":5088},[72,537],[28,5090,4936],{"className":5091},[72,537],[28,5093,5094,5097],{"style":4448},[28,5095],{"className":5096,"style":1773},[529],[28,5098,5099],{},[28,5100,4997],{"className":5101},[5057],[28,5103,571],{"className":5104},[570],[28,5106,5108],{"className":5107},[517],[28,5109,5112],{"className":5110,"style":5111},[521],"height:0.8104em;",[28,5113],{},[28,5115],{"className":5116,"style":135},[134],[28,5118,5120,5123],{"className":5119},[72],[28,5121,50],{"className":5122},[72],[28,5124,5126],{"className":5125},[508],[28,5127,5129],{"className":5128},[512],[28,5130,5132],{"className":5131},[517],[28,5133,5135],{"className":5134,"style":1690},[521],[28,5136,5137,5140],{"style":1357},[28,5138],{"className":5139,"style":530},[529],[28,5141,5143],{"className":5142},[534,535,536,537],[28,5144,469],{"className":5145},[72,503,537],[28,5147],{"className":5148,"style":810},[134],[28,5150,93],{"className":5151},[814],[28,5153],{"className":5154,"style":810},[134],[28,5156,5158,5161,5164,5167,5196,5199,5202],{"className":5157},[63],[28,5159],{"className":5160,"style":3703},[67],[28,5162,370],{"className":5163},[120],[28,5165,194],{"className":5166},[72],[28,5168,5170,5173],{"className":5169},[142],[28,5171,318],{"className":5172},[142],[28,5174,5176],{"className":5175},[508],[28,5177,5179],{"className":5178},[512],[28,5180,5182],{"className":5181},[517],[28,5183,5185],{"className":5184,"style":1690},[521],[28,5186,5187,5190],{"style":1357},[28,5188],{"className":5189,"style":530},[529],[28,5191,5193],{"className":5192},[534,535,536,537],[28,5194,469],{"className":5195},[72,503,537],[28,5197],{"className":5198,"style":584},[134],[28,5200,472],{"className":5201},[563],[28,5203],{"className":5204,"style":584},[134],[28,5206,5208,5212,5215,5218,5221],{"className":5207},[63],[28,5209],{"className":5210,"style":5211},[67],"height:0.7278em;vertical-align:-0.0833em;",[28,5213,50],{"className":5214},[72],[28,5216],{"className":5217,"style":810},[134],[28,5219,93],{"className":5220},[814],[28,5222],{"className":5223,"style":810},[134],[28,5225,5227,5230,5233,5236,5239],{"className":5226},[63],[28,5228],{"className":5229,"style":68},[67],[28,5231,1061],{"className":5232},[72],[28,5234],{"className":5235,"style":584},[134],[28,5237,472],{"className":5238},[563],[28,5240],{"className":5241,"style":584},[134],[28,5243,5245,5248],{"className":5244},[63],[28,5246],{"className":5247,"style":68},[67],[28,5249,50],{"className":5250},[72],[10,5252,5253,5254,5282],{},"That is to say, random and uniform samples tend to distribute on the outer shell as the dimension ",[28,5255,5257,5270],{"className":5256,"translate":32},[31],[28,5258,5260],{"className":5259},[36],[38,5261,5262],{"xmlns":40},[42,5263,5264,5268],{},[45,5265,5266],{},[458,5267,469],{},[52,5269,469],{"encoding":54},[28,5271,5273],{"className":5272,"ariaHidden":59},[58],[28,5274,5276,5279],{"className":5275},[63],[28,5277],{"className":5278,"style":3585},[67],[28,5280,469],{"className":5281},[72,503]," grows. This is named the Thin-Shell Effect.",[5284,5285,5287],"h3",{"id":5286},"euclidean-distance","Euclidean Distance",[10,5289,5290,5291,5353],{},"Consider a high-dimensional space ",[28,5292,5294,5314],{"className":5293,"translate":32},[31],[28,5295,5297],{"className":5296},[36],[38,5298,5299],{"xmlns":40},[42,5300,5301,5311],{},[45,5302,5303],{},[656,5304,5305,5309],{},[458,5306,5308],{"mathvariant":5307},"double-struck","R",[458,5310,469],{},[52,5312,5313],{"encoding":54},"\\mathbb{R}^d",[28,5315,5317],{"className":5316,"ariaHidden":59},[58],[28,5318,5320,5323],{"className":5319},[63],[28,5321],{"className":5322,"style":522},[67],[28,5324,5326,5330],{"className":5325},[72],[28,5327,5308],{"className":5328},[72,5329],"mathbb",[28,5331,5333],{"className":5332},[508],[28,5334,5336],{"className":5335},[512],[28,5337,5339],{"className":5338},[517],[28,5340,5342],{"className":5341,"style":522},[521],[28,5343,5344,5347],{"style":547},[28,5345],{"className":5346,"style":530},[529],[28,5348,5350],{"className":5349},[534,535,536,537],[28,5351,469],{"className":5352},[72,503,537]," with random vectors:",[28,5355,5357],{"className":5356,"translate":32},[1034],[28,5358,5360,5403],{"className":5359,"translate":32},[31],[28,5361,5363],{"className":5362},[36],[38,5364,5365],{"xmlns":40,"display":1043},[42,5366,5367,5400],{},[45,5368,5369,5371,5373,5375,5378,5381,5385,5387,5389,5391,5398],{},[458,5370,747],{},[87,5372,98],{"separator":59},[458,5374,757],{},[87,5376,5377],{},"∼",[462,5379,5380],{},"i.i.d. ",[458,5382,5384],{"mathvariant":5383},"script","N",[87,5386,370],{"stretchy":89},[48,5388,1061],{},[87,5390,98],{"separator":59},[1065,5392,5393,5396],{},[458,5394,5395],{},"I",[458,5397,469],{},[87,5399,318],{"stretchy":89},[52,5401,5402],{"encoding":54},"x, y \\sim \\text{i.i.d.} \\ \\mathcal{N}(0, I_d)",[28,5404,5406,5434],{"className":5405,"ariaHidden":59},[58],[28,5407,5409,5413,5416,5419,5422,5425,5428,5431],{"className":5408},[63],[28,5410],{"className":5411,"style":5412},[67],"height:0.625em;vertical-align:-0.1944em;",[28,5414,747],{"className":5415},[72,503],[28,5417,98],{"className":5418},[130],[28,5420],{"className":5421,"style":135},[134],[28,5423,757],{"className":5424,"style":699},[72,503],[28,5426],{"className":5427,"style":584},[134],[28,5429,5377],{"className":5430},[563],[28,5432],{"className":5433,"style":584},[134],[28,5435,5437,5440,5447,5451,5456,5459,5462,5465,5468,5510],{"className":5436},[63],[28,5438],{"className":5439,"style":116},[67],[28,5441,5443],{"className":5442},[72,541],[28,5444,5446],{"className":5445},[72],"i.i.d.",[28,5448,5450],{"className":5449},[134]," ",[28,5452,5384],{"className":5453,"style":5455},[72,5454],"mathcal","margin-right:0.14736em;",[28,5457,370],{"className":5458},[120],[28,5460,1061],{"className":5461},[72],[28,5463,98],{"className":5464},[130],[28,5466],{"className":5467,"style":135},[134],[28,5469,5471,5475],{"className":5470},[72],[28,5472,5395],{"className":5473,"style":5474},[72,503],"margin-right:0.07847em;",[28,5476,5478],{"className":5477},[508],[28,5479,5481,5502],{"className":5480},[512,513],[28,5482,5484,5499],{"className":5483},[517],[28,5485,5487],{"className":5486,"style":1409},[521],[28,5488,5490,5493],{"style":5489},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[28,5491],{"className":5492,"style":530},[529],[28,5494,5496],{"className":5495},[534,535,536,537],[28,5497,469],{"className":5498},[72,503,537],[28,5500,571],{"className":5501},[570],[28,5503,5505],{"className":5504},[517],[28,5506,5508],{"className":5507,"style":1306},[521],[28,5509],{},[28,5511,318],{"className":5512},[142],[10,5514,5515,5516,3619],{},"The magnitude of the vector ",[28,5517,5519,5532],{"className":5518,"translate":32},[31],[28,5520,5522],{"className":5521},[36],[38,5523,5524],{"xmlns":40},[42,5525,5526,5530],{},[45,5527,5528],{},[458,5529,747],{},[52,5531,747],{"encoding":54},[28,5533,5535],{"className":5534,"ariaHidden":59},[58],[28,5536,5538,5541],{"className":5537},[63],[28,5539],{"className":5540,"style":3615},[67],[28,5542,747],{"className":5543},[72,503],[28,5545,5547],{"className":5546,"translate":32},[1034],[28,5548,5550,5604],{"className":5549,"translate":32},[31],[28,5551,5553],{"className":5552},[36],[38,5554,5555],{"xmlns":40,"display":1043},[42,5556,5557,5601],{},[45,5558,5559,5562,5564,5566,5568,5574,5576,5593],{},[458,5560,5561],{"mathvariant":482},"∣",[458,5563,5561],{"mathvariant":482},[458,5565,747],{},[458,5567,5561],{"mathvariant":482},[656,5569,5570,5572],{},[458,5571,5561],{"mathvariant":482},[48,5573,619],{},[87,5575,472],{},[5577,5578,5579,5582,5591],"munderover",{},[87,5580,5581],{},"∑",[45,5583,5584,5587,5589],{},[458,5585,5586],{},"i",[87,5588,472],{},[48,5590,50],{},[458,5592,469],{},[455,5594,5595,5597,5599],{},[458,5596,747],{},[458,5598,5586],{},[48,5600,619],{},[52,5602,5603],{"encoding":54},"||x||^2 = \\sum_{i=1}^d x_i^2",[28,5605,5607,5662],{"className":5606,"ariaHidden":59},[58],[28,5608,5610,5614,5618,5621,5624,5653,5656,5659],{"className":5609},[63],[28,5611],{"className":5612,"style":5613},[67],"height:1.1141em;vertical-align:-0.25em;",[28,5615,5617],{"className":5616},[72],"∣∣",[28,5619,747],{"className":5620},[72,503],[28,5622,5561],{"className":5623},[72],[28,5625,5627,5630],{"className":5626},[72],[28,5628,5561],{"className":5629},[72],[28,5631,5633],{"className":5632},[508],[28,5634,5636],{"className":5635},[512],[28,5637,5639],{"className":5638},[517],[28,5640,5642],{"className":5641,"style":1354},[521],[28,5643,5644,5647],{"style":1357},[28,5645],{"className":5646,"style":530},[529],[28,5648,5650],{"className":5649},[534,535,536,537],[28,5651,619],{"className":5652},[72,537],[28,5654],{"className":5655,"style":584},[134],[28,5657,472],{"className":5658},[563],[28,5660],{"className":5661,"style":584},[134],[28,5663,5665,5669,5741,5744],{"className":5664},[63],[28,5666],{"className":5667,"style":5668},[67],"height:3.1138em;vertical-align:-1.2777em;",[28,5670,5672],{"className":5671},[5057,5058],[28,5673,5675,5732],{"className":5674},[512,513],[28,5676,5678,5729],{"className":5677},[517],[28,5679,5682,5704,5717],{"className":5680,"style":5681},[521],"height:1.8361em;",[28,5683,5685,5689],{"style":5684},"top:-1.8723em;margin-left:0em;",[28,5686],{"className":5687,"style":5688},[529],"height:3.05em;",[28,5690,5692],{"className":5691},[534,535,536,537],[28,5693,5695,5698,5701],{"className":5694},[72,537],[28,5696,5586],{"className":5697},[72,503,537],[28,5699,472],{"className":5700},[563,537],[28,5702,50],{"className":5703},[72,537],[28,5705,5707,5710],{"style":5706},"top:-3.05em;",[28,5708],{"className":5709,"style":5688},[529],[28,5711,5712],{},[28,5713,5581],{"className":5714},[5057,5715,5716],"op-symbol","large-op",[28,5718,5720,5723],{"style":5719},"top:-4.3em;margin-left:0em;",[28,5721],{"className":5722,"style":5688},[529],[28,5724,5726],{"className":5725},[534,535,536,537],[28,5727,469],{"className":5728},[72,503,537],[28,5730,571],{"className":5731},[570],[28,5733,5735],{"className":5734},[517],[28,5736,5739],{"className":5737,"style":5738},[521],"height:1.2777em;",[28,5740],{},[28,5742],{"className":5743,"style":135},[134],[28,5745,5747,5750],{"className":5746},[72],[28,5748,747],{"className":5749},[72,503],[28,5751,5753],{"className":5752},[508],[28,5754,5756,5788],{"className":5755},[512,513],[28,5757,5759,5785],{"className":5758},[517],[28,5760,5762,5774],{"className":5761,"style":1354},[521],[28,5763,5765,5768],{"style":5764},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[28,5766],{"className":5767,"style":530},[529],[28,5769,5771],{"className":5770},[534,535,536,537],[28,5772,5586],{"className":5773},[72,503,537],[28,5775,5776,5779],{"style":1357},[28,5777],{"className":5778,"style":530},[529],[28,5780,5782],{"className":5781},[534,535,536,537],[28,5783,619],{"className":5784},[72,537],[28,5786,571],{"className":5787},[570],[28,5789,5791],{"className":5790},[517],[28,5792,5794],{"className":5793,"style":1735},[521],[28,5795],{},[10,5797,5798,5799,5954],{},"Since ",[28,5800,5802,5837],{"className":5801,"translate":32},[31],[28,5803,5805],{"className":5804},[36],[38,5806,5807],{"xmlns":40},[42,5808,5809,5834],{},[45,5810,5811,5819,5821,5828,5830,5832],{},[455,5812,5813,5815,5817],{},[458,5814,747],{},[458,5816,5586],{},[48,5818,619],{},[87,5820,5377],{},[656,5822,5823,5826],{},[458,5824,5825],{},"χ",[48,5827,619],{},[87,5829,370],{"stretchy":89},[48,5831,50],{},[87,5833,318],{"stretchy":89},[52,5835,5836],{"encoding":54},"x_i^2 \\sim \\chi^2(1)",[28,5838,5840,5909],{"className":5839,"ariaHidden":59},[58],[28,5841,5843,5847,5900,5903,5906],{"className":5842},[63],[28,5844],{"className":5845,"style":5846},[67],"height:1.0728em;vertical-align:-0.2587em;",[28,5848,5850,5853],{"className":5849},[72],[28,5851,747],{"className":5852},[72,503],[28,5854,5856],{"className":5855},[508],[28,5857,5859,5891],{"className":5858},[512,513],[28,5860,5862,5888],{"className":5861},[517],[28,5863,5865,5877],{"className":5864,"style":695},[521],[28,5866,5868,5871],{"style":5867},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[28,5869],{"className":5870,"style":530},[529],[28,5872,5874],{"className":5873},[534,535,536,537],[28,5875,5586],{"className":5876},[72,503,537],[28,5878,5879,5882],{"style":547},[28,5880],{"className":5881,"style":530},[529],[28,5883,5885],{"className":5884},[534,535,536,537],[28,5886,619],{"className":5887},[72,537],[28,5889,571],{"className":5890},[570],[28,5892,5894],{"className":5893},[517],[28,5895,5898],{"className":5896,"style":5897},[521],"height:0.2587em;",[28,5899],{},[28,5901],{"className":5902,"style":584},[134],[28,5904,5377],{"className":5905},[563],[28,5907],{"className":5908,"style":584},[134],[28,5910,5912,5916,5945,5948,5951],{"className":5911},[63],[28,5913],{"className":5914,"style":5915},[67],"height:1.0641em;vertical-align:-0.25em;",[28,5917,5919,5922],{"className":5918},[72],[28,5920,5825],{"className":5921},[72,503],[28,5923,5925],{"className":5924},[508],[28,5926,5928],{"className":5927},[512],[28,5929,5931],{"className":5930},[517],[28,5932,5934],{"className":5933,"style":695},[521],[28,5935,5936,5939],{"style":547},[28,5937],{"className":5938,"style":530},[529],[28,5940,5942],{"className":5941},[534,535,536,537],[28,5943,619],{"className":5944},[72,537],[28,5946,370],{"className":5947},[120],[28,5949,50],{"className":5950},[72],[28,5952,318],{"className":5953},[142],", we have:",[28,5956,5958],{"className":5957,"translate":32},[1034],[28,5959,5961,6001],{"className":5960,"translate":32},[31],[28,5962,5964],{"className":5963},[36],[38,5965,5966],{"xmlns":40,"display":1043},[42,5967,5968,5998],{},[45,5969,5970,5972,5974,5976,5978,5984,5986,5992,5994,5996],{},[458,5971,5561],{"mathvariant":482},[458,5973,5561],{"mathvariant":482},[458,5975,747],{},[458,5977,5561],{"mathvariant":482},[656,5979,5980,5982],{},[458,5981,5561],{"mathvariant":482},[48,5983,619],{},[87,5985,5377],{},[656,5987,5988,5990],{},[458,5989,5825],{},[48,5991,619],{},[87,5993,370],{"stretchy":89},[458,5995,469],{},[87,5997,318],{"stretchy":89},[52,5999,6000],{"encoding":54},"||x||^2 \\sim \\chi^2(d)",[28,6002,6004,6057],{"className":6003,"ariaHidden":59},[58],[28,6005,6007,6010,6013,6016,6019,6048,6051,6054],{"className":6006},[63],[28,6008],{"className":6009,"style":5613},[67],[28,6011,5617],{"className":6012},[72],[28,6014,747],{"className":6015},[72,503],[28,6017,5561],{"className":6018},[72],[28,6020,6022,6025],{"className":6021},[72],[28,6023,5561],{"className":6024},[72],[28,6026,6028],{"className":6027},[508],[28,6029,6031],{"className":6030},[512],[28,6032,6034],{"className":6033},[517],[28,6035,6037],{"className":6036,"style":1354},[521],[28,6038,6039,6042],{"style":1357},[28,6040],{"className":6041,"style":530},[529],[28,6043,6045],{"className":6044},[534,535,536,537],[28,6046,619],{"className":6047},[72,537],[28,6049],{"className":6050,"style":584},[134],[28,6052,5377],{"className":6053},[563],[28,6055],{"className":6056,"style":584},[134],[28,6058,6060,6063,6092,6095,6098],{"className":6059},[63],[28,6061],{"className":6062,"style":5613},[67],[28,6064,6066,6069],{"className":6065},[72],[28,6067,5825],{"className":6068},[72,503],[28,6070,6072],{"className":6071},[508],[28,6073,6075],{"className":6074},[512],[28,6076,6078],{"className":6077},[517],[28,6079,6081],{"className":6080,"style":1354},[521],[28,6082,6083,6086],{"style":1357},[28,6084],{"className":6085,"style":530},[529],[28,6087,6089],{"className":6088},[534,535,536,537],[28,6090,619],{"className":6091},[72,537],[28,6093,370],{"className":6094},[120],[28,6096,469],{"className":6097},[72,503],[28,6099,318],{"className":6100},[142],[10,6102,6103,6104,6180,6181,6209,6210,6267,6268,6296,6297,409],{},"Here, ",[28,6105,6107,6132],{"className":6106,"translate":32},[31],[28,6108,6110],{"className":6109},[36],[38,6111,6112],{"xmlns":40},[42,6113,6114,6129],{},[45,6115,6116,6122,6124,6127],{},[656,6117,6118,6120],{},[458,6119,5825],{},[48,6121,619],{},[87,6123,370],{"stretchy":89},[458,6125,6126],{},"k",[87,6128,318],{"stretchy":89},[52,6130,6131],{"encoding":54},"\\chi^2(k)",[28,6133,6135],{"className":6134,"ariaHidden":59},[58],[28,6136,6138,6141,6170,6173,6177],{"className":6137},[63],[28,6139],{"className":6140,"style":5915},[67],[28,6142,6144,6147],{"className":6143},[72],[28,6145,5825],{"className":6146},[72,503],[28,6148,6150],{"className":6149},[508],[28,6151,6153],{"className":6152},[512],[28,6154,6156],{"className":6155},[517],[28,6157,6159],{"className":6158,"style":695},[521],[28,6160,6161,6164],{"style":547},[28,6162],{"className":6163,"style":530},[529],[28,6165,6167],{"className":6166},[534,535,536,537],[28,6168,619],{"className":6169},[72,537],[28,6171,370],{"className":6172},[120],[28,6174,6126],{"className":6175,"style":6176},[72,503],"margin-right:0.03148em;",[28,6178,318],{"className":6179},[142]," is the Chi-Squared Distribution, defined as the sum of squares of ",[28,6182,6184,6197],{"className":6183,"translate":32},[31],[28,6185,6187],{"className":6186},[36],[38,6188,6189],{"xmlns":40},[42,6190,6191,6195],{},[45,6192,6193],{},[458,6194,6126],{},[52,6196,6126],{"encoding":54},[28,6198,6200],{"className":6199,"ariaHidden":59},[58],[28,6201,6203,6206],{"className":6202},[63],[28,6204],{"className":6205,"style":3585},[67],[28,6207,6126],{"className":6208,"style":6176},[72,503]," mutually independent random variables following a standard normal distribution ",[28,6211,6213,6237],{"className":6212,"translate":32},[31],[28,6214,6216],{"className":6215},[36],[38,6217,6218],{"xmlns":40},[42,6219,6220,6234],{},[45,6221,6222,6224,6226,6228,6230,6232],{},[458,6223,5384],{"mathvariant":5383},[87,6225,370],{"stretchy":89},[48,6227,1061],{},[87,6229,98],{"separator":59},[48,6231,50],{},[87,6233,318],{"stretchy":89},[52,6235,6236],{"encoding":54},"\\mathcal{N}(0,1)",[28,6238,6240],{"className":6239,"ariaHidden":59},[58],[28,6241,6243,6246,6249,6252,6255,6258,6261,6264],{"className":6242},[63],[28,6244],{"className":6245,"style":116},[67],[28,6247,5384],{"className":6248,"style":5455},[72,5454],[28,6250,370],{"className":6251},[120],[28,6253,1061],{"className":6254},[72],[28,6256,98],{"className":6257},[130],[28,6259],{"className":6260,"style":135},[134],[28,6262,50],{"className":6263},[72],[28,6265,318],{"className":6266},[142],". In this case, since the squared norm is the sum of ",[28,6269,6271,6284],{"className":6270,"translate":32},[31],[28,6272,6274],{"className":6273},[36],[38,6275,6276],{"xmlns":40},[42,6277,6278,6282],{},[45,6279,6280],{},[458,6281,469],{},[52,6283,469],{"encoding":54},[28,6285,6287],{"className":6286,"ariaHidden":59},[58],[28,6288,6290,6293],{"className":6289},[63],[28,6291],{"className":6292,"style":3585},[67],[28,6294,469],{"className":6295},[72,503]," such variables, the degrees of freedom is ",[28,6298,6300,6318],{"className":6299,"translate":32},[31],[28,6301,6303],{"className":6302},[36],[38,6304,6305],{"xmlns":40},[42,6306,6307,6315],{},[45,6308,6309,6311,6313],{},[458,6310,6126],{},[87,6312,472],{},[458,6314,469],{},[52,6316,6317],{"encoding":54},"k=d",[28,6319,6321,6339],{"className":6320,"ariaHidden":59},[58],[28,6322,6324,6327,6330,6333,6336],{"className":6323},[63],[28,6325],{"className":6326,"style":3585},[67],[28,6328,6126],{"className":6329,"style":6176},[72,503],[28,6331],{"className":6332,"style":584},[134],[28,6334,472],{"className":6335},[563],[28,6337],{"className":6338,"style":584},[134],[28,6340,6342,6345],{"className":6341},[63],[28,6343],{"className":6344,"style":3585},[67],[28,6346,469],{"className":6347},[72,503],[10,6349,6350],{},"According to the Law of Large Numbers:",[28,6352,6354],{"className":6353,"translate":32},[1034],[28,6355,6357,6393],{"className":6356,"translate":32},[31],[28,6358,6360],{"className":6359},[36],[38,6361,6362],{"xmlns":40,"display":1043},[42,6363,6364,6390],{},[45,6365,6366,6372,6374,6376,6378,6380,6386,6388],{},[1187,6367,6368,6370],{},[48,6369,50],{},[458,6371,469],{},[458,6373,5561],{"mathvariant":482},[458,6375,5561],{"mathvariant":482},[458,6377,747],{},[458,6379,5561],{"mathvariant":482},[656,6381,6382,6384],{},[458,6383,5561],{"mathvariant":482},[48,6385,619],{},[87,6387,3459],{},[48,6389,50],{},[52,6391,6392],{"encoding":54},"\\frac{1}{d}||x||^2 \\to 1",[28,6394,6396,6512],{"className":6395,"ariaHidden":59},[58],[28,6397,6399,6403,6465,6468,6471,6474,6503,6506,6509],{"className":6398},[63],[28,6400],{"className":6401,"style":6402},[67],"height:2.0074em;vertical-align:-0.686em;",[28,6404,6406,6409,6462],{"className":6405},[72],[28,6407],{"className":6408},[120,1753],[28,6410,6412],{"className":6411},[1187],[28,6413,6415,6454],{"className":6414},[512,513],[28,6416,6418,6451],{"className":6417},[517],[28,6419,6421,6432,6440],{"className":6420,"style":1925},[521],[28,6422,6423,6426],{"style":1769},[28,6424],{"className":6425,"style":1773},[529],[28,6427,6429],{"className":6428},[72],[28,6430,469],{"className":6431},[72,503],[28,6433,6434,6437],{"style":1822},[28,6435],{"className":6436,"style":1773},[529],[28,6438],{"className":6439,"style":1830},[1829],[28,6441,6442,6445],{"style":1833},[28,6443],{"className":6444,"style":1773},[529],[28,6446,6448],{"className":6447},[72],[28,6449,50],{"className":6450},[72],[28,6452,571],{"className":6453},[570],[28,6455,6457],{"className":6456},[517],[28,6458,6460],{"className":6459,"style":1968},[521],[28,6461],{},[28,6463],{"className":6464},[142,1753],[28,6466,5617],{"className":6467},[72],[28,6469,747],{"className":6470},[72,503],[28,6472,5561],{"className":6473},[72],[28,6475,6477,6480],{"className":6476},[72],[28,6478,5561],{"className":6479},[72],[28,6481,6483],{"className":6482},[508],[28,6484,6486],{"className":6485},[512],[28,6487,6489],{"className":6488},[517],[28,6490,6492],{"className":6491,"style":1354},[521],[28,6493,6494,6497],{"style":1357},[28,6495],{"className":6496,"style":530},[529],[28,6498,6500],{"className":6499},[534,535,536,537],[28,6501,619],{"className":6502},[72,537],[28,6504],{"className":6505,"style":584},[134],[28,6507,3459],{"className":6508},[563],[28,6510],{"className":6511,"style":584},[134],[28,6513,6515,6518],{"className":6514},[63],[28,6516],{"className":6517,"style":68},[67],[28,6519,50],{"className":6520},[72],[10,6522,6523,6524,409],{},"That is, the magnitude of vectors is approximated by ",[28,6525,6527,6556],{"className":6526,"translate":32},[31],[28,6528,6530],{"className":6529},[36],[38,6531,6532],{"xmlns":40},[42,6533,6534,6553],{},[45,6535,6536,6538,6540,6542,6544,6546,6549],{},[458,6537,5561],{"mathvariant":482},[458,6539,5561],{"mathvariant":482},[458,6541,747],{},[458,6543,5561],{"mathvariant":482},[458,6545,5561],{"mathvariant":482},[87,6547,6548],{},"≈",[4413,6550,6551],{},[458,6552,469],{},[52,6554,6555],{"encoding":54},"||x|| \\approx 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have the Euclidean 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0",[28,7593,7595,7699],{"className":7594,"ariaHidden":59},[58],[28,7596,7598,7601,7604,7607,7647,7687,7690,7693,7696],{"className":7597},[63],[28,7599],{"className":7600,"style":116},[67],[28,7602,7568],{"className":7603,"style":679},[72,503],[28,7605,90],{"className":7606},[120],[28,7608,7610,7613],{"className":7609},[72],[28,7611,747],{"className":7612},[72,503],[28,7614,7616],{"className":7615},[508],[28,7617,7619,7639],{"className":7618},[512,513],[28,7620,7622,7636],{"className":7621},[517],[28,7623,7625],{"className":7624,"style":7355},[521],[28,7626,7627,7630],{"style":7358},[28,7628],{"className":7629,"style":530},[529],[28,7631,7633],{"className":7632},[534,535,536,537],[28,7634,5586],{"className":7635},[72,503,537],[28,7637,571],{"className":7638},[570],[28,7640,7642],{"className":7641},[517],[28,7643,7645],{"className":7644,"style":1306},[521],[28,7646],{},[28,7648,7650,7653],{"className":7649},[72],[28,7651,757],{"className":7652,"style":699},[72,503],[28,7654,7656],{"className":7655},[508],[28,7657,7659,7679],{"className":7658},[512,513],[28,7660,7662,7676],{"className":7661},[517],[28,7663,7665],{"className":7664,"style":7355},[521],[28,7666,7667,7670],{"style":7399},[28,7668],{"className":7669,"style":530},[529],[28,7671,7673],{"className":7672},[534,535,536,537],[28,7674,5586],{"className":7675},[72,503,537],[28,7677,571],{"className":7678},[570],[28,7680,7682],{"className":7681},[517],[28,7683,7685],{"className":7684,"style":1306},[521],[28,7686],{},[28,7688,103],{"className":7689},[142],[28,7691],{"className":7692,"style":584},[134],[28,7694,472],{"className":7695},[563],[28,7697],{"className":7698,"style":584},[134],[28,7700,7702,7705],{"className":7701},[63],[28,7703],{"className":7704,"style":68},[67],[28,7706,1061],{"className":7707},[72],[3997,7709,7710],{},[28,7711,7713,7748],{"className":7712,"translate":32},[31],[28,7714,7716],{"className":7715},[36],[38,7717,7718],{"xmlns":40},[42,7719,7720,7745],{},[45,7721,7722,7725,7727,7733,7739,7741,7743],{},[462,7723,7724],{},"Var",[87,7726,370],{"stretchy":89},[1065,7728,7729,7731],{},[458,7730,747],{},[458,7732,5586],{},[1065,7734,7735,7737],{},[458,7736,757],{},[458,7738,5586],{},[87,7740,318],{"stretchy":89},[87,7742,472],{},[48,7744,50],{},[52,7746,7747],{"encoding":54},"\\text{Var}(x_i y_i) = 1",[28,7749,7751,7858],{"className":7750,"ariaHidden":59},[58],[28,7752,7754,7757,7763,7766,7806,7846,7849,7852,7855],{"className":7753},[63],[28,7755],{"className":7756,"style":116},[67],[28,7758,7760],{"className":7759},[72,541],[28,7761,7724],{"className":7762},[72],[28,7764,370],{"className":7765},[120],[28,7767,7769,7772],{"className":7768},[72],[28,7770,747],{"className":7771},[72,503],[28,7773,7775],{"className":7774},[508],[28,7776,7778,7798],{"className":7777},[512,513],[28,7779,7781,7795],{"className":7780},[517],[28,7782,7784],{"className":7783,"style":7355},[521],[28,7785,7786,7789],{"style":7358},[28,7787],{"className":7788,"style":530},[529],[28,7790,7792],{"className":7791},[534,535,536,537],[28,7793,5586],{"className":7794},[72,503,537],[28,7796,571],{"className":7797},[570],[28,7799,7801],{"className":7800},[517],[28,7802,7804],{"className":7803,"style":1306},[521],[28,7805],{},[28,7807,7809,7812],{"className":7808},[72],[28,7810,757],{"className":7811,"style":699},[72,503],[28,7813,7815],{"className":7814},[508],[28,7816,7818,7838],{"className":7817},[512,513],[28,7819,7821,7835],{"className":7820},[517],[28,7822,7824],{"className":7823,"style":7355},[521],[28,7825,7826,7829],{"style":7399},[28,7827],{"className":7828,"style":530},[529],[28,7830,7832],{"className":7831},[534,535,536,537],[28,7833,5586],{"className":7834},[72,503,537],[28,7836,571],{"className":7837},[570],[28,7839,7841],{"className":7840},[517],[28,7842,7844],{"className":7843,"style":1306},[521],[28,7845],{},[28,7847,318],{"className":7848},[142],[28,7850],{"className":7851,"style":584},[134],[28,7853,472],{"className":7854},[563],[28,7856],{"className":7857,"style":584},[134],[28,7859,7861,7864],{"className":7860},[63],[28,7862],{"className":7863,"style":68},[67],[28,7865,50],{"className":7866},[72],[10,7868,7869],{},"Therefore:",[28,7871,7873],{"className":7872,"translate":32},[1034],[28,7874,7876,7912],{"className":7875,"translate":32},[31],[28,7877,7879],{"className":7878},[36],[38,7880,7881],{"xmlns":40,"display":1043},[42,7882,7883,7909],{},[45,7884,7885,7887,7889,7891,7893,7895,7897,7899,7901,7903,7905,7907],{},[87,7886,6714],{"stretchy":89},[458,7888,747],{},[87,7890,98],{"separator":59},[458,7892,757],{},[87,7894,6723],{"stretchy":89},[87,7896,5377],{},[458,7898,5384],{"mathvariant":5383},[87,7900,370],{"stretchy":89},[48,7902,1061],{},[87,7904,98],{"separator":59},[458,7906,469],{},[87,7908,318],{"stretchy":89},[52,7910,7911],{"encoding":54},"\\langle x, y \\rangle \\sim \\mathcal{N}(0, d)",[28,7913,7915,7948],{"className":7914,"ariaHidden":59},[58],[28,7916,7918,7921,7924,7927,7930,7933,7936,7939,7942,7945],{"className":7917},[63],[28,7919],{"className":7920,"style":116},[67],[28,7922,6714],{"className":7923},[120],[28,7925,747],{"className":7926},[72,503],[28,7928,98],{"className":7929},[130],[28,7931],{"className":7932,"style":135},[134],[28,7934,757],{"className":7935,"style":699},[72,503],[28,7937,6723],{"className":7938},[142],[28,7940],{"className":7941,"style":584},[134],[28,7943,5377],{"className":7944},[563],[28,7946],{"className":7947,"style":584},[134],[28,7949,7951,7954,7957,7960,7963,7966,7969,7972],{"className":7950},[63],[28,7952],{"className":7953,"style":116},[67],[28,7955,5384],{"className":7956,"style":5455},[72,5454],[28,7958,370],{"className":7959},[120],[28,7961,1061],{"className":7962},[72],[28,7964,98],{"className":7965},[130],[28,7967],{"className":7968,"style":135},[134],[28,7970,469],{"className":7971},[72,503],[28,7973,318],{"className":7974},[142],[10,7976,7977,7978,409],{},"The order of magnitude is ",[28,7979,7981,8004],{"className":7980,"translate":32},[31],[28,7982,7984],{"className":7983},[36],[38,7985,7986],{"xmlns":40},[42,7987,7988,8001],{},[45,7989,7990,7993,7995,7999],{},[458,7991,7992],{},"O",[87,7994,370],{"stretchy":89},[4413,7996,7997],{},[458,7998,469],{},[87,8000,318],{"stretchy":89},[52,8002,8003],{"encoding":54},"O(\\sqrt{d})",[28,8005,8007],{"className":8006,"ariaHidden":59},[58],[28,8008,8010,8014,8017,8020,8067],{"className":8009},[63],[28,8011],{"className":8012,"style":8013},[67],"height:1.1822em;vertical-align:-0.25em;",[28,8015,7992],{"className":8016,"style":706},[72,503],[28,8018,370],{"className":8019},[120],[28,8021,8023],{"className":8022},[72,4433],[28,8024,8026,8059],{"className":8025},[512,513],[28,8027,8029,8056],{"className":8028},[517],[28,8030,8032,8044],{"className":8031,"style":6602},[521],[28,8033,8035,8038],{"className":8034,"style":4448},[4447],[28,8036],{"className":8037,"style":1773},[529],[28,8039,8041],{"className":8040,"style":4455},[72],[28,8042,469],{"className":8043},[72,503],[28,8045,8046,8049],{"style":6617},[28,8047],{"className":8048,"style":1773},[529],[28,8050,8052],{"className":8051,"style":4469},[4468],[4471,8053,8054],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,8055],{"d":4481},[28,8057,571],{"className":8058},[570],[28,8060,8062],{"className":8061},[517],[28,8063,8065],{"className":8064,"style":6637},[521],[28,8066],{},[28,8068,318],{"className":8069},[142],[10,8071,8072],{},"Substituting back into the distance equation, we have:",[28,8074,8076],{"className":8075,"translate":32},[1034],[28,8077,8079,8178],{"className":8078,"translate":32},[31],[28,8080,8082],{"className":8081},[36],[38,8083,8084],{"xmlns":40,"display":1043},[42,8085,8086,8175],{},[45,8087,8088,8090,8092,8094,8096,8098,8100,8106,8108,8110,8112,8114,8116,8118,8120,8122,8124,8128,8130,8132,8134,8136,8138,8140,8142,8146,8148,8151,8153,8155,8157,8159,8161,8163,8169,8171,8173],{},[458,8089,5561],{"mathvariant":482},[458,8091,5561],{"mathvariant":482},[458,8093,747],{},[87,8095,93],{},[458,8097,757],{},[458,8099,5561],{"mathvariant":482},[656,8101,8102,8104],{},[458,8103,5561],{"mathvariant":482},[48,8105,619],{},[87,8107,472],{},[458,8109,469],{},[87,8111,752],{},[458,8113,469],{},[87,8115,93],{},[48,8117,619],{},[87,8119,1078],{},[458,8121,7992],{},[87,8123,370],{"stretchy":89},[4413,8125,8126],{},[458,8127,469],{},[87,8129,318],{"stretchy":89},[87,8131,472],{},[48,8133,619],{},[458,8135,469],{},[87,8137,752],{},[458,8139,7992],{},[87,8141,370],{"stretchy":89},[4413,8143,8144],{},[458,8145,469],{},[87,8147,318],{"stretchy":89},[134,8149],{"linebreak":8150},"newline",[458,8152,5561],{"mathvariant":482},[458,8154,5561],{"mathvariant":482},[458,8156,747],{},[87,8158,93],{},[458,8160,757],{},[458,8162,5561],{"mathvariant":482},[656,8164,8165,8167],{},[458,8166,5561],{"mathvariant":482},[48,8168,619],{},[87,8170,6548],{},[48,8172,619],{},[458,8174,469],{},[52,8176,8177],{"encoding":54},"||x - y||^2\n= d + d - 2 \\cdot O(\\sqrt{d})\n= 2d + O(\\sqrt{d}) \\\\\n||x - y||^2 \\approx 2d",[28,8179,8181,8202,8252,8271,8289,8307,8382,8403,8465,8468,8489,8539],{"className":8180,"ariaHidden":59},[58],[28,8182,8184,8187,8190,8193,8196,8199],{"className":8183},[63],[28,8185],{"className":8186,"style":116},[67],[28,8188,5617],{"className":8189},[72],[28,8191,747],{"className":8192},[72,503],[28,8194],{"className":8195,"style":810},[134],[28,8197,93],{"className":8198},[814],[28,8200],{"className":8201,"style":810},[134],[28,8203,8205,8208,8211,8214,8243,8246,8249],{"className":8204},[63],[28,8206],{"className":8207,"style":5613},[67],[28,8209,757],{"className":8210,"style":699},[72,503],[28,8212,5561],{"className":8213},[72],[28,8215,8217,8220],{"className":8216},[72],[28,8218,5561],{"className":8219},[72],[28,8221,8223],{"className":8222},[508],[28,8224,8226],{"className":8225},[512],[28,8227,8229],{"className":8228},[517],[28,8230,8232],{"className":8231,"style":1354},[521],[28,8233,8234,8237],{"style":1357},[28,8235],{"className":8236,"style":530},[529],[28,8238,8240],{"className":8239},[534,535,536,537],[28,8241,619],{"className":8242},[72,537],[28,8244],{"className":8245,"style":584},[134],[28,8247,472],{"className":8248},[563],[28,8250],{"className":8251,"style":584},[134],[28,8253,8255,8259,8262,8265,8268],{"className":8254},[63],[28,8256],{"className":8257,"style":8258},[67],"height:0.7778em;vertical-align:-0.0833em;",[28,8260,469],{"className":8261},[72,503],[28,8263],{"className":8264,"style":810},[134],[28,8266,752],{"className":8267},[814],[28,8269],{"className":8270,"style":810},[134],[28,8272,8274,8277,8280,8283,8286],{"className":8273},[63],[28,8275],{"className":8276,"style":8258},[67],[28,8278,469],{"className":8279},[72,503],[28,8281],{"className":8282,"style":810},[134],[28,8284,93],{"className":8285},[814],[28,8287],{"className":8288,"style":810},[134],[28,8290,8292,8295,8298,8301,8304],{"className":8291},[63],[28,8293],{"className":8294,"style":68},[67],[28,8296,619],{"className":8297},[72],[28,8299],{"className":8300,"style":810},[134],[28,8302,1078],{"className":8303},[814],[28,8305],{"className":8306,"style":810},[134],[28,8308,8310,8314,8317,8320,8370,8373,8376,8379],{"className":8309},[63],[28,8311],{"className":8312,"style":8313},[67],"height:1.2311em;vertical-align:-0.25em;",[28,8315,7992],{"className":8316,"style":706},[72,503],[28,8318,370],{"className":8319},[120],[28,8321,8323],{"className":8322},[72,4433],[28,8324,8326,8361],{"className":8325},[512,513],[28,8327,8329,8358],{"className":8328},[517],[28,8330,8333,8345],{"className":8331,"style":8332},[521],"height:0.9811em;",[28,8334,8336,8339],{"className":8335,"style":4448},[4447],[28,8337],{"className":8338,"style":1773},[529],[28,8340,8342],{"className":8341,"style":4455},[72],[28,8343,469],{"className":8344},[72,503],[28,8346,8348,8351],{"style":8347},"top:-2.9411em;",[28,8349],{"className":8350,"style":1773},[529],[28,8352,8354],{"className":8353,"style":4469},[4468],[4471,8355,8356],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,8357],{"d":4481},[28,8359,571],{"className":8360},[570],[28,8362,8364],{"className":8363},[517],[28,8365,8368],{"className":8366,"style":8367},[521],"height:0.0589em;",[28,8369],{},[28,8371,318],{"className":8372},[142],[28,8374],{"className":8375,"style":584},[134],[28,8377,472],{"className":8378},[563],[28,8380],{"className":8381,"style":584},[134],[28,8383,8385,8388,8391,8394,8397,8400],{"className":8384},[63],[28,8386],{"className":8387,"style":8258},[67],[28,8389,619],{"className":8390},[72],[28,8392,469],{"className":8393},[72,503],[28,8395],{"className":8396,"style":810},[134],[28,8398,752],{"className":8399},[814],[28,8401],{"className":8402,"style":810},[134],[28,8404,8406,8409,8412,8415,8462],{"className":8405},[63],[28,8407],{"className":8408,"style":8313},[67],[28,8410,7992],{"className":8411,"style":706},[72,503],[28,8413,370],{"className":8414},[120],[28,8416,8418],{"className":8417},[72,4433],[28,8419,8421,8454],{"className":8420},[512,513],[28,8422,8424,8451],{"className":8423},[517],[28,8425,8427,8439],{"className":8426,"style":8332},[521],[28,8428,8430,8433],{"className":8429,"style":4448},[4447],[28,8431],{"className":8432,"style":1773},[529],[28,8434,8436],{"className":8435,"style":4455},[72],[28,8437,469],{"className":8438},[72,503],[28,8440,8441,8444],{"style":8347},[28,8442],{"className":8443,"style":1773},[529],[28,8445,8447],{"className":8446,"style":4469},[4468],[4471,8448,8449],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,8450],{"d":4481},[28,8452,571],{"className":8453},[570],[28,8455,8457],{"className":8456},[517],[28,8458,8460],{"className":8459,"style":8367},[521],[28,8461],{},[28,8463,318],{"className":8464},[142],[28,8466],{"className":8467},[134,8150],[28,8469,8471,8474,8477,8480,8483,8486],{"className":8470},[63],[28,8472],{"className":8473,"style":116},[67],[28,8475,5617],{"className":8476},[72],[28,8478,747],{"className":8479},[72,503],[28,8481],{"className":8482,"style":810},[134],[28,8484,93],{"className":8485},[814],[28,8487],{"className":8488,"style":810},[134],[28,8490,8492,8495,8498,8501,8530,8533,8536],{"className":8491},[63],[28,8493],{"className":8494,"style":5613},[67],[28,8496,757],{"className":8497,"style":699},[72,503],[28,8499,5561],{"className":8500},[72],[28,8502,8504,8507],{"className":8503},[72],[28,8505,5561],{"className":8506},[72],[28,8508,8510],{"className":8509},[508],[28,8511,8513],{"className":8512},[512],[28,8514,8516],{"className":8515},[517],[28,8517,8519],{"className":8518,"style":1354},[521],[28,8520,8521,8524],{"style":1357},[28,8522],{"className":8523,"style":530},[529],[28,8525,8527],{"className":8526},[534,535,536,537],[28,8528,619],{"className":8529},[72,537],[28,8531],{"className":8532,"style":584},[134],[28,8534,6548],{"className":8535},[563],[28,8537],{"className":8538,"style":584},[134],[28,8540,8542,8545,8548],{"className":8541},[63],[28,8543],{"className":8544,"style":3585},[67],[28,8546,619],{"className":8547},[72],[28,8549,469],{"className":8550},[72,503],[10,8552,8553],{},"Thus:",[28,8555,8557],{"className":8556,"translate":32},[1034],[28,8558,8560,8596],{"className":8559,"translate":32},[31],[28,8561,8563],{"className":8562},[36],[38,8564,8565],{"xmlns":40,"display":1043},[42,8566,8567,8593],{},[45,8568,8569,8571,8573,8575,8577,8579,8581,8583,8585],{},[458,8570,5561],{"mathvariant":482},[458,8572,5561],{"mathvariant":482},[458,8574,747],{},[87,8576,93],{},[458,8578,757],{},[458,8580,5561],{"mathvariant":482},[458,8582,5561],{"mathvariant":482},[87,8584,6548],{},[4413,8586,8587],{},[45,8588,8589,8591],{},[48,8590,619],{},[458,8592,469],{},[52,8594,8595],{"encoding":54},"||x - y|| \\approx \\sqrt{2d}",[28,8597,8599,8620,8641],{"className":8598,"ariaHidden":59},[58],[28,8600,8602,8605,8608,8611,8614,8617],{"className":8601},[63],[28,8603],{"className":8604,"style":116},[67],[28,8606,5617],{"className":8607},[72],[28,8609,747],{"className":8610},[72,503],[28,8612],{"className":8613,"style":810},[134],[28,8615,93],{"className":8616},[814],[28,8618],{"className":8619,"style":810},[134],[28,8621,8623,8626,8629,8632,8635,8638],{"className":8622},[63],[28,8624],{"className":8625,"style":116},[67],[28,8627,757],{"className":8628,"style":699},[72,503],[28,8630,5617],{"className":8631},[72],[28,8633],{"className":8634,"style":584},[134],[28,8636,6548],{"className":8637},[563],[28,8639],{"className":8640,"style":584},[134],[28,8642,8644,8648],{"className":8643},[63],[28,8645],{"className":8646,"style":8647},[67],"height:1.04em;vertical-align:-0.0589em;",[28,8649,8651],{"className":8650},[72,4433],[28,8652,8654,8690],{"className":8653},[512,513],[28,8655,8657,8687],{"className":8656},[517],[28,8658,8660,8675],{"className":8659,"style":8332},[521],[28,8661,8663,8666],{"className":8662,"style":4448},[4447],[28,8664],{"className":8665,"style":1773},[529],[28,8667,8669,8672],{"className":8668,"style":4455},[72],[28,8670,619],{"className":8671},[72],[28,8673,469],{"className":8674},[72,503],[28,8676,8677,8680],{"style":8347},[28,8678],{"className":8679,"style":1773},[529],[28,8681,8683],{"className":8682,"style":4469},[4468],[4471,8684,8685],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,8686],{"d":4481},[28,8688,571],{"className":8689},[570],[28,8691,8693],{"className":8692},[517],[28,8694,8696],{"className":8695,"style":8367},[521],[28,8697],{},[10,8699,8700],{},"Looking at the relative fluctuation:",[3994,8702,8703,8740],{},[3997,8704,8705,8706],{},"Mean: ",[28,8707,8709,8725],{"className":8708,"translate":32},[31],[28,8710,8712],{"className":8711},[36],[38,8713,8714],{"xmlns":40},[42,8715,8716,8722],{},[45,8717,8718,8720],{},[48,8719,619],{},[458,8721,469],{},[52,8723,8724],{"encoding":54},"2d",[28,8726,8728],{"className":8727,"ariaHidden":59},[58],[28,8729,8731,8734,8737],{"className":8730},[63],[28,8732],{"className":8733,"style":3585},[67],[28,8735,619],{"className":8736},[72],[28,8738,469],{"className":8739},[72,503],[3997,8741,8742,8743],{},"Standard Deviation: ",[28,8744,8746,8767],{"className":8745,"translate":32},[31],[28,8747,8749],{"className":8748},[36],[38,8750,8751],{"xmlns":40},[42,8752,8753,8765],{},[45,8754,8755,8757,8759,8763],{},[458,8756,7992],{},[87,8758,370],{"stretchy":89},[4413,8760,8761],{},[458,8762,469],{},[87,8764,318],{"stretchy":89},[52,8766,8003],{"encoding":54},[28,8768,8770],{"className":8769,"ariaHidden":59},[58],[28,8771,8773,8776,8779,8782,8829],{"className":8772},[63],[28,8774],{"className":8775,"style":8013},[67],[28,8777,7992],{"className":8778,"style":706},[72,503],[28,8780,370],{"className":8781},[120],[28,8783,8785],{"className":8784},[72,4433],[28,8786,8788,8821],{"className":8787},[512,513],[28,8789,8791,8818],{"className":8790},[517],[28,8792,8794,8806],{"className":8793,"style":6602},[521],[28,8795,8797,8800],{"className":8796,"style":4448},[4447],[28,8798],{"className":8799,"style":1773},[529],[28,8801,8803],{"className":8802,"style":4455},[72],[28,8804,469],{"className":8805},[72,503],[28,8807,8808,8811],{"style":6617},[28,8809],{"className":8810,"style":1773},[529],[28,8812,8814],{"className":8813,"style":4469},[4468],[4471,8815,8816],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,8817],{"d":4481},[28,8819,571],{"className":8820},[570],[28,8822,8824],{"className":8823},[517],[28,8825,8827],{"className":8826,"style":6637},[521],[28,8828],{},[28,8830,318],{"className":8831},[142],[10,8833,7869],{},[28,8835,8837],{"className":8836,"translate":32},[1034],[28,8838,8840,8884],{"className":8839,"translate":32},[31],[28,8841,8843],{"className":8842},[36],[38,8844,8845],{"xmlns":40,"display":1043},[42,8846,8847,8881],{},[45,8848,8849,8857,8859,8867,8869,8877,8879],{},[1187,8850,8851,8854],{},[462,8852,8853],{},"std",[462,8855,8856],{},"mean",[87,8858,5377],{},[1187,8860,8861,8865],{},[4413,8862,8863],{},[458,8864,469],{},[458,8866,469],{},[87,8868,472],{},[1187,8870,8871,8873],{},[48,8872,50],{},[4413,8874,8875],{},[458,8876,469],{},[87,8878,3459],{},[48,8880,1061],{},[52,8882,8883],{"encoding":54},"\\frac{\\text{std}}{\\text{mean}} \\sim \\frac{\\sqrt{d}}{d} = \\frac{1}{\\sqrt{d}} \\to 0",[28,8885,8887,8972,9095,9219],{"className":8886,"ariaHidden":59},[58],[28,8888,8890,8894,8963,8966,8969],{"className":8889},[63],[28,8891],{"className":8892,"style":8893},[67],"height:2.0574em;vertical-align:-0.686em;",[28,8895,8897,8900,8960],{"className":8896},[72],[28,8898],{"className":8899},[120,1753],[28,8901,8903],{"className":8902},[1187],[28,8904,8906,8952],{"className":8905},[512,513],[28,8907,8909,8949],{"className":8908},[517],[28,8910,8913,8927,8935],{"className":8911,"style":8912},[521],"height:1.3714em;",[28,8914,8915,8918],{"style":1769},[28,8916],{"className":8917,"style":1773},[529],[28,8919,8921],{"className":8920},[72],[28,8922,8924],{"className":8923},[72,541],[28,8925,8856],{"className":8926},[72],[28,8928,8929,8932],{"style":1822},[28,8930],{"className":8931,"style":1773},[529],[28,8933],{"className":8934,"style":1830},[1829],[28,8936,8937,8940],{"style":1833},[28,8938],{"className":8939,"style":1773},[529],[28,8941,8943],{"className":8942},[72],[28,8944,8946],{"className":8945},[72,541],[28,8947,8853],{"className":8948},[72],[28,8950,571],{"className":8951},[570],[28,8953,8955],{"className":8954},[517],[28,8956,8958],{"className":8957,"style":1968},[521],[28,8959],{},[28,8961],{"className":8962},[142,1753],[28,8964],{"className":8965,"style":584},[134],[28,8967,5377],{"className":8968},[563],[28,8970],{"className":8971,"style":584},[134],[28,8973,8975,8979,9086,9089,9092],{"className":8974},[63],[28,8976],{"className":8977,"style":8978},[67],"height:2.2952em;vertical-align:-0.686em;",[28,8980,8982,8985,9083],{"className":8981},[72],[28,8983],{"className":8984},[120,1753],[28,8986,8988],{"className":8987},[1187],[28,8989,8991,9075],{"className":8990},[512,513],[28,8992,8994,9072],{"className":8993},[517],[28,8995,8998,9009,9017],{"className":8996,"style":8997},[521],"height:1.6092em;",[28,8999,9000,9003],{"style":1769},[28,9001],{"className":9002,"style":1773},[529],[28,9004,9006],{"className":9005},[72],[28,9007,469],{"className":9008},[72,503],[28,9010,9011,9014],{"style":1822},[28,9012],{"className":9013,"style":1773},[529],[28,9015],{"className":9016,"style":1830},[1829],[28,9018,9019,9022],{"style":1833},[28,9020],{"className":9021,"style":1773},[529],[28,9023,9025],{"className":9024},[72],[28,9026,9028],{"className":9027},[72,4433],[28,9029,9031,9064],{"className":9030},[512,513],[28,9032,9034,9061],{"className":9033},[517],[28,9035,9037,9049],{"className":9036,"style":6602},[521],[28,9038,9040,9043],{"className":9039,"style":4448},[4447],[28,9041],{"className":9042,"style":1773},[529],[28,9044,9046],{"className":9045,"style":4455},[72],[28,9047,469],{"className":9048},[72,503],[28,9050,9051,9054],{"style":6617},[28,9052],{"className":9053,"style":1773},[529],[28,9055,9057],{"className":9056,"style":4469},[4468],[4471,9058,9059],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,9060],{"d":4481},[28,9062,571],{"className":9063},[570],[28,9065,9067],{"className":9066},[517],[28,9068,9070],{"className":9069,"style":6637},[521],[28,9071],{},[28,9073,571],{"className":9074},[570],[28,9076,9078],{"className":9077},[517],[28,9079,9081],{"className":9080,"style":1968},[521],[28,9082],{},[28,9084],{"className":9085},[142,1753],[28,9087],{"className":9088,"style":584},[134],[28,9090,472],{"className":9091},[563],[28,9093],{"className":9094,"style":584},[134],[28,9096,9098,9102,9210,9213,9216],{"className":9097},[63],[28,9099],{"className":9100,"style":9101},[67],"height:2.2514em;vertical-align:-0.93em;",[28,9103,9105,9108,9207],{"className":9104},[72],[28,9106],{"className":9107},[120,1753],[28,9109,9111],{"className":9110},[1187],[28,9112,9114,9198],{"className":9113},[512,513],[28,9115,9117,9195],{"className":9116},[517],[28,9118,9120,9176,9184],{"className":9119,"style":1925},[521],[28,9121,9123,9126],{"style":9122},"top:-2.1778em;",[28,9124],{"className":9125,"style":1773},[529],[28,9127,9129],{"className":9128},[72],[28,9130,9132],{"className":9131},[72,4433],[28,9133,9135,9168],{"className":9134},[512,513],[28,9136,9138,9165],{"className":9137},[517],[28,9139,9141,9153],{"className":9140,"style":6602},[521],[28,9142,9144,9147],{"className":9143,"style":4448},[4447],[28,9145],{"className":9146,"style":1773},[529],[28,9148,9150],{"className":9149,"style":4455},[72],[28,9151,469],{"className":9152},[72,503],[28,9154,9155,9158],{"style":6617},[28,9156],{"className":9157,"style":1773},[529],[28,9159,9161],{"className":9160,"style":4469},[4468],[4471,9162,9163],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,9164],{"d":4481},[28,9166,571],{"className":9167},[570],[28,9169,9171],{"className":9170},[517],[28,9172,9174],{"className":9173,"style":6637},[521],[28,9175],{},[28,9177,9178,9181],{"style":1822},[28,9179],{"className":9180,"style":1773},[529],[28,9182],{"className":9183,"style":1830},[1829],[28,9185,9186,9189],{"style":1833},[28,9187],{"className":9188,"style":1773},[529],[28,9190,9192],{"className":9191},[72],[28,9193,50],{"className":9194},[72],[28,9196,571],{"className":9197},[570],[28,9199,9201],{"className":9200},[517],[28,9202,9205],{"className":9203,"style":9204},[521],"height:0.93em;",[28,9206],{},[28,9208],{"className":9209},[142,1753],[28,9211],{"className":9212,"style":584},[134],[28,9214,3459],{"className":9215},[563],[28,9217],{"className":9218,"style":584},[134],[28,9220,9222,9225],{"className":9221},[63],[28,9223],{"className":9224,"style":68},[67],[28,9226,1061],{"className":9227},[72],[10,9229,9230,9231,9261],{},"Conclusion: The relative fluctuation of distance tends to 0 ",[28,9232,9234,9248],{"className":9233,"translate":32},[31],[28,9235,9237],{"className":9236},[36],[38,9238,9239],{"xmlns":40},[42,9240,9241,9245],{},[45,9242,9243],{},[87,9244,3459],{},[52,9246,9247],{"encoding":54},"\\rightarrow",[28,9249,9251],{"className":9250,"ariaHidden":59},[58],[28,9252,9254,9258],{"className":9253},[63],[28,9255],{"className":9256,"style":9257},[67],"height:0.3669em;",[28,9259,3459],{"className":9260},[563]," the distance between all points is almost the same.",[5284,9263,9265],{"id":9264},"angular-distance","Angular Distance",[10,9267,9268,9269,9342],{},"If we project vectors onto the unit hypersphere after ",[28,9270,9272,9291],{"className":9271,"translate":32},[31],[28,9273,9275],{"className":9274},[36],[38,9276,9277],{"xmlns":40},[42,9278,9279,9288],{},[45,9280,9281],{},[1065,9282,9283,9286],{},[458,9284,9285],{"mathvariant":482},"ℓ",[48,9287,619],{},[52,9289,9290],{"encoding":54},"\\ell_2",[28,9292,9294],{"className":9293,"ariaHidden":59},[58],[28,9295,9297,9301],{"className":9296},[63],[28,9298],{"className":9299,"style":9300},[67],"height:0.8444em;vertical-align:-0.15em;",[28,9302,9304,9307],{"className":9303},[72],[28,9305,9285],{"className":9306},[72],[28,9308,9310],{"className":9309},[508],[28,9311,9313,9334],{"className":9312},[512,513],[28,9314,9316,9331],{"className":9315},[517],[28,9317,9320],{"className":9318,"style":9319},[521],"height:0.3011em;",[28,9321,9322,9325],{"style":7358},[28,9323],{"className":9324,"style":530},[529],[28,9326,9328],{"className":9327},[534,535,536,537],[28,9329,619],{"className":9330},[72,537],[28,9332,571],{"className":9333},[570],[28,9335,9337],{"className":9336},[517],[28,9338,9340],{"className":9339,"style":1306},[521],[28,9341],{}," normalization:",[28,9344,9346],{"className":9345,"translate":32},[1034],[28,9347,9349,9387],{"className":9348,"translate":32},[31],[28,9350,9352],{"className":9351},[36],[38,9353,9354],{"xmlns":40,"display":1043},[42,9355,9356,9384],{},[45,9357,9358,9360,9362,9364,9366,9368,9370,9372,9374,9376,9378,9380,9382],{},[458,9359,5561],{"mathvariant":482},[458,9361,5561],{"mathvariant":482},[458,9363,747],{},[458,9365,5561],{"mathvariant":482},[458,9367,5561],{"mathvariant":482},[87,9369,472],{},[458,9371,5561],{"mathvariant":482},[458,9373,5561],{"mathvariant":482},[458,9375,757],{},[458,9377,5561],{"mathvariant":482},[458,9379,5561],{"mathvariant":482},[87,9381,472],{},[48,9383,50],{},[52,9385,9386],{"encoding":54},"||x|| = ||y|| = 1",[28,9388,9390,9414,9438],{"className":9389,"ariaHidden":59},[58],[28,9391,9393,9396,9399,9402,9405,9408,9411],{"className":9392},[63],[28,9394],{"className":9395,"style":116},[67],[28,9397,5617],{"className":9398},[72],[28,9400,747],{"className":9401},[72,503],[28,9403,5617],{"className":9404},[72],[28,9406],{"className":9407,"style":584},[134],[28,9409,472],{"className":9410},[563],[28,9412],{"className":9413,"style":584},[134],[28,9415,9417,9420,9423,9426,9429,9432,9435],{"className":9416},[63],[28,9418],{"className":9419,"style":116},[67],[28,9421,5617],{"className":9422},[72],[28,9424,757],{"className":9425,"style":699},[72,503],[28,9427,5617],{"className":9428},[72],[28,9430],{"className":9431,"style":584},[134],[28,9433,472],{"className":9434},[563],[28,9436],{"className":9437,"style":584},[134],[28,9439,9441,9444],{"className":9440},[63],[28,9442],{"className":9443,"style":68},[67],[28,9445,50],{"className":9446},[72],[10,9448,9449],{},"Then:",[28,9451,9453],{"className":9452,"translate":32},[1034],[28,9454,9456,9502],{"className":9455,"translate":32},[31],[28,9457,9459],{"className":9458},[36],[38,9460,9461],{"xmlns":40,"display":1043},[42,9462,9463,9499],{},[45,9464,9465,9467,9469,9471,9473,9475,9477,9483,9485,9487,9489,9491,9494,9496],{},[458,9466,5561],{"mathvariant":482},[458,9468,5561],{"mathvariant":482},[458,9470,747],{},[87,9472,93],{},[458,9474,757],{},[458,9476,5561],{"mathvariant":482},[656,9478,9479,9481],{},[458,9480,5561],{"mathvariant":482},[48,9482,619],{},[87,9484,472],{},[48,9486,619],{},[87,9488,93],{},[48,9490,619],{},[458,9492,9493],{},"cos",[87,9495,5000],{},[458,9497,9498],{},"θ",[52,9500,9501],{"encoding":54},"||x - y||^2 = 2 - 2\\cos\\theta",[28,9503,9505,9526,9576,9594],{"className":9504,"ariaHidden":59},[58],[28,9506,9508,9511,9514,9517,9520,9523],{"className":9507},[63],[28,9509],{"className":9510,"style":116},[67],[28,9512,5617],{"className":9513},[72],[28,9515,747],{"className":9516},[72,503],[28,9518],{"className":9519,"style":810},[134],[28,9521,93],{"className":9522},[814],[28,9524],{"className":9525,"style":810},[134],[28,9527,9529,9532,9535,9538,9567,9570,9573],{"className":9528},[63],[28,9530],{"className":9531,"style":5613},[67],[28,9533,757],{"className":9534,"style":699},[72,503],[28,9536,5561],{"className":9537},[72],[28,9539,9541,9544],{"className":9540},[72],[28,9542,5561],{"className":9543},[72],[28,9545,9547],{"className":9546},[508],[28,9548,9550],{"className":9549},[512],[28,9551,9553],{"className":9552},[517],[28,9554,9556],{"className":9555,"style":1354},[521],[28,9557,9558,9561],{"style":1357},[28,9559],{"className":9560,"style":530},[529],[28,9562,9564],{"className":9563},[534,535,536,537],[28,9565,619],{"className":9566},[72,537],[28,9568],{"className":9569,"style":584},[134],[28,9571,472],{"className":9572},[563],[28,9574],{"className":9575,"style":584},[134],[28,9577,9579,9582,9585,9588,9591],{"className":9578},[63],[28,9580],{"className":9581,"style":5211},[67],[28,9583,619],{"className":9584},[72],[28,9586],{"className":9587,"style":810},[134],[28,9589,93],{"className":9590},[814],[28,9592],{"className":9593,"style":810},[134],[28,9595,9597,9600,9603,9606,9609,9612],{"className":9596},[63],[28,9598],{"className":9599,"style":3585},[67],[28,9601,619],{"className":9602},[72],[28,9604],{"className":9605,"style":135},[134],[28,9607,9493],{"className":9608},[5057],[28,9610],{"className":9611,"style":135},[134],[28,9613,9498],{"className":9614,"style":706},[72,503],[10,9616,9617],{},"In high dimensions:",[28,9619,9621],{"className":9620,"translate":32},[1034],[28,9622,9624,9646],{"className":9623,"translate":32},[31],[28,9625,9627],{"className":9626},[36],[38,9628,9629],{"xmlns":40,"display":1043},[42,9630,9631,9643],{},[45,9632,9633,9635,9637,9639,9641],{},[458,9634,9493],{},[87,9636,5000],{},[458,9638,9498],{},[87,9640,6548],{},[48,9642,1061],{},[52,9644,9645],{"encoding":54},"\\cos\\theta \\approx 0",[28,9647,9649,9673],{"className":9648,"ariaHidden":59},[58],[28,9650,9652,9655,9658,9661,9664,9667,9670],{"className":9651},[63],[28,9653],{"className":9654,"style":3585},[67],[28,9656,9493],{"className":9657},[5057],[28,9659],{"className":9660,"style":135},[134],[28,9662,9498],{"className":9663,"style":706},[72,503],[28,9665],{"className":9666,"style":584},[134],[28,9668,6548],{"className":9669},[563],[28,9671],{"className":9672,"style":584},[134],[28,9674,9676,9679],{"className":9675},[63],[28,9677],{"className":9678,"style":68},[67],[28,9680,1061],{"className":9681},[72],[10,9683,7869],{},[28,9685,9687],{"className":9686,"translate":32},[1034],[28,9688,9690,9722],{"className":9689,"translate":32},[31],[28,9691,9693],{"className":9692},[36],[38,9694,9695],{"xmlns":40,"display":1043},[42,9696,9697,9719],{},[45,9698,9699,9701,9703,9705,9707,9709,9711,9713,9715],{},[458,9700,5561],{"mathvariant":482},[458,9702,5561],{"mathvariant":482},[458,9704,747],{},[87,9706,93],{},[458,9708,757],{},[458,9710,5561],{"mathvariant":482},[458,9712,5561],{"mathvariant":482},[87,9714,6548],{},[4413,9716,9717],{},[48,9718,619],{},[52,9720,9721],{"encoding":54},"||x - y|| \\approx \\sqrt{2}",[28,9723,9725,9746,9767],{"className":9724,"ariaHidden":59},[58],[28,9726,9728,9731,9734,9737,9740,9743],{"className":9727},[63],[28,9729],{"className":9730,"style":116},[67],[28,9732,5617],{"className":9733},[72],[28,9735,747],{"className":9736},[72,503],[28,9738],{"className":9739,"style":810},[134],[28,9741,93],{"className":9742},[814],[28,9744],{"className":9745,"style":810},[134],[28,9747,9749,9752,9755,9758,9761,9764],{"className":9748},[63],[28,9750],{"className":9751,"style":116},[67],[28,9753,757],{"className":9754,"style":699},[72,503],[28,9756,5617],{"className":9757},[72],[28,9759],{"className":9760,"style":584},[134],[28,9762,6548],{"className":9763},[563],[28,9765],{"className":9766,"style":584},[134],[28,9768,9770,9774],{"className":9769},[63],[28,9771],{"className":9772,"style":9773},[67],"height:1.04em;vertical-align:-0.0839em;",[28,9775,9777],{"className":9776},[72,4433],[28,9778,9780,9815],{"className":9779},[512,513],[28,9781,9783,9812],{"className":9782},[517],[28,9784,9787,9799],{"className":9785,"style":9786},[521],"height:0.9561em;",[28,9788,9790,9793],{"className":9789,"style":4448},[4447],[28,9791],{"className":9792,"style":1773},[529],[28,9794,9796],{"className":9795,"style":4455},[72],[28,9797,619],{"className":9798},[72],[28,9800,9802,9805],{"style":9801},"top:-2.9161em;",[28,9803],{"className":9804,"style":1773},[529],[28,9806,9808],{"className":9807,"style":4469},[4468],[4471,9809,9810],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,9811],{"d":4481},[28,9813,571],{"className":9814},[570],[28,9816,9818],{"className":9817},[517],[28,9819,9822],{"className":9820,"style":9821},[521],"height:0.0839em;",[28,9823],{},[10,9825,9826],{},"The angular distance between all vectors is almost equidistant.",[10,9828,9829,9830,9919,9920,9964],{},"In high-dimensional spaces, the norm of random vectors concentrates around a constant due to the law of large numbers. Moreover, the inner product between independent vectors grows only on the order of ",[28,9831,9833,9854],{"className":9832,"translate":32},[31],[28,9834,9836],{"className":9835},[36],[38,9837,9838],{"xmlns":40},[42,9839,9840,9852],{},[45,9841,9842,9844,9846,9850],{},[458,9843,7992],{},[87,9845,370],{"stretchy":89},[4413,9847,9848],{},[458,9849,469],{},[87,9851,318],{"stretchy":89},[52,9853,8003],{"encoding":54},[28,9855,9857],{"className":9856,"ariaHidden":59},[58],[28,9858,9860,9863,9866,9869,9916],{"className":9859},[63],[28,9861],{"className":9862,"style":8013},[67],[28,9864,7992],{"className":9865,"style":706},[72,503],[28,9867,370],{"className":9868},[120],[28,9870,9872],{"className":9871},[72,4433],[28,9873,9875,9908],{"className":9874},[512,513],[28,9876,9878,9905],{"className":9877},[517],[28,9879,9881,9893],{"className":9880,"style":6602},[521],[28,9882,9884,9887],{"className":9883,"style":4448},[4447],[28,9885],{"className":9886,"style":1773},[529],[28,9888,9890],{"className":9889,"style":4455},[72],[28,9891,469],{"className":9892},[72,503],[28,9894,9895,9898],{"style":6617},[28,9896],{"className":9897,"style":1773},[529],[28,9899,9901],{"className":9900,"style":4469},[4468],[4471,9902,9903],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,9904],{"d":4481},[28,9906,571],{"className":9907},[570],[28,9909,9911],{"className":9910},[517],[28,9912,9914],{"className":9913,"style":6637},[521],[28,9915],{},[28,9917,318],{"className":9918},[142],", which is negligible compared to the ",[28,9921,9923,9943],{"className":9922,"translate":32},[31],[28,9924,9926],{"className":9925},[36],[38,9927,9928],{"xmlns":40},[42,9929,9930,9940],{},[45,9931,9932,9934,9936,9938],{},[458,9933,7992],{},[87,9935,370],{"stretchy":89},[458,9937,469],{},[87,9939,318],{"stretchy":89},[52,9941,9942],{"encoding":54},"O(d)",[28,9944,9946],{"className":9945,"ariaHidden":59},[58],[28,9947,9949,9952,9955,9958,9961],{"className":9948},[63],[28,9950],{"className":9951,"style":116},[67],[28,9953,7992],{"className":9954,"style":706},[72,503],[28,9956,370],{"className":9957},[120],[28,9959,469],{"className":9960},[72,503],[28,9962,318],{"className":9963},[142]," magnitude of squared norms. As a result, pairwise distances concentrate around a constant, leading to the phenomenon that almost all points are approximately equidistant.",[17,9966,9968],{"id":9967},"positively-curved-riemannian-space","Positively Curved Riemannian Space",[10,9970,9971],{},"In Euclidean space, the curse of dimensionality arises from the polynomial growth of volume and the independence structure of coordinates. However, in a positively curved Riemannian manifold, the geometry itself fundamentally reshapes both volume distribution and distance behavior.",[10,9973,9974,9975,10003],{},"A canonical example is the ",[28,9976,9978,9991],{"className":9977,"translate":32},[31],[28,9979,9981],{"className":9980},[36],[38,9982,9983],{"xmlns":40},[42,9984,9985,9989],{},[45,9986,9987],{},[458,9988,469],{},[52,9990,469],{"encoding":54},[28,9992,9994],{"className":9993,"ariaHidden":59},[58],[28,9995,9997,10000],{"className":9996},[63],[28,9998],{"className":9999,"style":3585},[67],[28,10001,469],{"className":10002},[72,503],"-dimensional unit sphere:",[28,10005,10007],{"className":10006,"translate":32},[1034],[28,10008,10010,10066],{"className":10009,"translate":32},[31],[28,10011,10013],{"className":10012},[36],[38,10014,10015],{"xmlns":40,"display":1043},[42,10016,10017,10063],{},[45,10018,10019,10031,10033],{},[656,10020,10021,10023],{},[458,10022,649],{"mathvariant":5307},[45,10024,10025,10027,10029],{},[458,10026,469],{},[87,10028,93],{},[48,10030,50],{},[87,10032,472],{},[45,10034,10035,10037,10040,10046,10049,10051,10053,10055,10057,10059,10061],{},[458,10036,747],{},[87,10038,10039],{},"∈",[656,10041,10042,10044],{},[458,10043,5308],{"mathvariant":5307},[458,10045,469],{},[87,10047,10048],{},":",[458,10050,5561],{"mathvariant":482},[458,10052,5561],{"mathvariant":482},[458,10054,747],{},[458,10056,5561],{"mathvariant":482},[458,10058,5561],{"mathvariant":482},[87,10060,472],{},[48,10062,50],{},[52,10064,10065],{"encoding":54},"\\mathbb{S}^{d-1} = { x \\in \\mathbb{R}^d : ||x|| = 1 }",[28,10067,10069,10122],{"className":10068,"ariaHidden":59},[58],[28,10070,10072,10075,10113,10116,10119],{"className":10071},[63],[28,10073],{"className":10074,"style":1690},[67],[28,10076,10078,10081],{"className":10077},[72],[28,10079,649],{"className":10080},[72,5329],[28,10082,10084],{"className":10083},[508],[28,10085,10087],{"className":10086},[512],[28,10088,10090],{"className":10089},[517],[28,10091,10093],{"className":10092,"style":1690},[521],[28,10094,10095,10098],{"style":1357},[28,10096],{"className":10097,"style":530},[529],[28,10099,10101],{"className":10100},[534,535,536,537],[28,10102,10104,10107,10110],{"className":10103},[72,537],[28,10105,469],{"className":10106},[72,503,537],[28,10108,93],{"className":10109},[814,537],[28,10111,50],{"className":10112},[72,537],[28,10114],{"className":10115,"style":584},[134],[28,10117,472],{"className":10118},[563],[28,10120],{"className":10121,"style":584},[134],[28,10123,10125,10128],{"className":10124},[63],[28,10126],{"className":10127,"style":3703},[67],[28,10129,10131,10134,10137,10140,10143,10172,10175,10178,10181,10184,10187,10190,10193,10196,10199],{"className":10130},[72],[28,10132,747],{"className":10133},[72,503],[28,10135],{"className":10136,"style":584},[134],[28,10138,10039],{"className":10139},[563],[28,10141],{"className":10142,"style":584},[134],[28,10144,10146,10149],{"className":10145},[72],[28,10147,5308],{"className":10148},[72,5329],[28,10150,10152],{"className":10151},[508],[28,10153,10155],{"className":10154},[512],[28,10156,10158],{"className":10157},[517],[28,10159,10161],{"className":10160,"style":1690},[521],[28,10162,10163,10166],{"style":1357},[28,10164],{"className":10165,"style":530},[529],[28,10167,10169],{"className":10168},[534,535,536,537],[28,10170,469],{"className":10171},[72,503,537],[28,10173],{"className":10174,"style":584},[134],[28,10176,10048],{"className":10177},[563],[28,10179],{"className":10180,"style":584},[134],[28,10182,5617],{"className":10183},[72],[28,10185,747],{"className":10186},[72,503],[28,10188,5617],{"className":10189},[72],[28,10191],{"className":10192,"style":584},[134],[28,10194,472],{"className":10195},[563],[28,10197],{"className":10198,"style":584},[134],[28,10200,50],{"className":10201},[72],[10,10203,10204],{},"which is a space with constant positive sectional curvature.",[5284,10206,10208],{"id":10207},"measure-concentration-on-the-sphere","Measure Concentration on the Sphere",[10,10210,10211],{},"Unlike Euclidean space, where volume spreads radially, on the sphere all points lie on a fixed-radius manifold. Thus, “radial shells” are replaced by geodesic bands.",[10,10213,10214,10215,10353],{},"Let us fix a point ",[28,10216,10218,10250],{"className":10217,"translate":32},[31],[28,10219,10221],{"className":10220},[36],[38,10222,10223],{"xmlns":40},[42,10224,10225,10247],{},[45,10226,10227,10233,10235],{},[1065,10228,10229,10231],{},[458,10230,747],{},[48,10232,1061],{},[87,10234,10039],{},[656,10236,10237,10239],{},[458,10238,649],{"mathvariant":5307},[45,10240,10241,10243,10245],{},[458,10242,469],{},[87,10244,93],{},[48,10246,50],{},[52,10248,10249],{"encoding":54},"x_0 \\in \\mathbb{S}^{d-1}",[28,10251,10253,10309],{"className":10252,"ariaHidden":59},[58],[28,10254,10256,10260,10300,10303,10306],{"className":10255},[63],[28,10257],{"className":10258,"style":10259},[67],"height:0.6891em;vertical-align:-0.15em;",[28,10261,10263,10266],{"className":10262},[72],[28,10264,747],{"className":10265},[72,503],[28,10267,10269],{"className":10268},[508],[28,10270,10272,10292],{"className":10271},[512,513],[28,10273,10275,10289],{"className":10274},[517],[28,10276,10278],{"className":10277,"style":9319},[521],[28,10279,10280,10283],{"style":7358},[28,10281],{"className":10282,"style":530},[529],[28,10284,10286],{"className":10285},[534,535,536,537],[28,10287,1061],{"className":10288},[72,537],[28,10290,571],{"className":10291},[570],[28,10293,10295],{"className":10294},[517],[28,10296,10298],{"className":10297,"style":1306},[521],[28,10299],{},[28,10301],{"className":10302,"style":584},[134],[28,10304,10039],{"className":10305},[563],[28,10307],{"className":10308,"style":584},[134],[28,10310,10312,10315],{"className":10311},[63],[28,10313],{"className":10314,"style":522},[67],[28,10316,10318,10321],{"className":10317},[72],[28,10319,649],{"className":10320},[72,5329],[28,10322,10324],{"className":10323},[508],[28,10325,10327],{"className":10326},[512],[28,10328,10330],{"className":10329},[517],[28,10331,10333],{"className":10332,"style":522},[521],[28,10334,10335,10338],{"style":547},[28,10336],{"className":10337,"style":530},[529],[28,10339,10341],{"className":10340},[534,535,536,537],[28,10342,10344,10347,10350],{"className":10343},[72,537],[28,10345,469],{"className":10346},[72,503,537],[28,10348,93],{"className":10349},[814,537],[28,10351,50],{"className":10352},[72,537]," and define a geodesic ball:",[28,10355,10357],{"className":10356,"translate":32},[1034],[28,10358,10360,10424],{"className":10359,"translate":32},[31],[28,10361,10363],{"className":10362},[36],[38,10364,10365],{"xmlns":40,"display":1043},[42,10366,10367,10421],{},[45,10368,10369,10372,10374,10376,10378,10380],{},[458,10370,10371],{},"B",[87,10373,370],{"stretchy":89},[458,10375,9498],{},[87,10377,318],{"stretchy":89},[87,10379,472],{},[45,10381,10382,10384,10386,10398,10400,10403,10405,10407,10409,10415,10417,10419],{},[458,10383,747],{},[87,10385,10039],{},[656,10387,10388,10390],{},[458,10389,649],{"mathvariant":5307},[45,10391,10392,10394,10396],{},[458,10393,469],{},[87,10395,93],{},[48,10397,50],{},[87,10399,10048],{},[458,10401,10402],{"mathvariant":482},"∠",[87,10404,370],{"stretchy":89},[458,10406,747],{},[87,10408,98],{"separator":59},[1065,10410,10411,10413],{},[458,10412,747],{},[48,10414,1061],{},[87,10416,318],{"stretchy":89},[87,10418,762],{},[458,10420,9498],{},[52,10422,10423],{"encoding":54},"B(\\theta) = { x \\in \\mathbb{S}^{d-1} : \\angle(x, x_0) \\leq \\theta }",[28,10425,10427,10455],{"className":10426,"ariaHidden":59},[58],[28,10428,10430,10433,10437,10440,10443,10446,10449,10452],{"className":10429},[63],[28,10431],{"className":10432,"style":116},[67],[28,10434,10371],{"className":10435,"style":10436},[72,503],"margin-right:0.05017em;",[28,10438,370],{"className":10439},[120],[28,10441,9498],{"className":10442,"style":706},[72,503],[28,10444,318],{"className":10445},[142],[28,10447],{"className":10448,"style":584},[134],[28,10450,472],{"className":10451},[563],[28,10453],{"className":10454,"style":584},[134],[28,10456,10458,10461],{"className":10457},[63],[28,10459],{"className":10460,"style":3703},[67],[28,10462,10464,10467,10470,10473,10476,10514,10517,10520,10523,10526,10529,10532,10535,10538,10578,10581,10584,10587,10590],{"className":10463},[72],[28,10465,747],{"className":10466},[72,503],[28,10468],{"className":10469,"style":584},[134],[28,10471,10039],{"className":10472},[563],[28,10474],{"className":10475,"style":584},[134],[28,10477,10479,10482],{"className":10478},[72],[28,10480,649],{"className":10481},[72,5329],[28,10483,10485],{"className":10484},[508],[28,10486,10488],{"className":10487},[512],[28,10489,10491],{"className":10490},[517],[28,10492,10494],{"className":10493,"style":1690},[521],[28,10495,10496,10499],{"style":1357},[28,10497],{"className":10498,"style":530},[529],[28,10500,10502],{"className":10501},[534,535,536,537],[28,10503,10505,10508,10511],{"className":10504},[72,537],[28,10506,469],{"className":10507},[72,503,537],[28,10509,93],{"className":10510},[814,537],[28,10512,50],{"className":10513},[72,537],[28,10515],{"className":10516,"style":584},[134],[28,10518,10048],{"className":10519},[563],[28,10521],{"className":10522,"style":584},[134],[28,10524,10402],{"className":10525},[72],[28,10527,370],{"className":10528},[120],[28,10530,747],{"className":10531},[72,503],[28,10533,98],{"className":10534},[130],[28,10536],{"className":10537,"style":135},[134],[28,10539,10541,10544],{"className":10540},[72],[28,10542,747],{"className":10543},[72,503],[28,10545,10547],{"className":10546},[508],[28,10548,10550,10570],{"className":10549},[512,513],[28,10551,10553,10567],{"className":10552},[517],[28,10554,10556],{"className":10555,"style":9319},[521],[28,10557,10558,10561],{"style":7358},[28,10559],{"className":10560,"style":530},[529],[28,10562,10564],{"className":10563},[534,535,536,537],[28,10565,1061],{"className":10566},[72,537],[28,10568,571],{"className":10569},[570],[28,10571,10573],{"className":10572},[517],[28,10574,10576],{"className":10575,"style":1306},[521],[28,10577],{},[28,10579,318],{"className":10580},[142],[28,10582],{"className":10583,"style":584},[134],[28,10585,762],{"className":10586},[563],[28,10588],{"className":10589,"style":584},[134],[28,10591,9498],{"className":10592,"style":706},[72,503],[10,10594,10595,10596,10048],{},"The measure of this region depends on ",[28,10597,10599,10634],{"className":10598,"translate":32},[31],[28,10600,10602],{"className":10601},[36],[38,10603,10604],{"xmlns":40},[42,10605,10606,10631],{},[45,10607,10608,10625,10627,10629],{},[656,10609,10610,10617],{},[45,10611,10612,10615],{},[458,10613,10614],{},"sin",[87,10616,5000],{},[45,10618,10619,10621,10623],{},[458,10620,469],{},[87,10622,93],{},[48,10624,619],{},[87,10626,370],{"stretchy":89},[458,10628,9498],{},[87,10630,318],{"stretchy":89},[52,10632,10633],{"encoding":54},"\\sin^{d-2}(\\theta)",[28,10635,10637],{"className":10636,"ariaHidden":59},[58],[28,10638,10640,10644,10684,10687,10690],{"className":10639},[63],[28,10641],{"className":10642,"style":10643},[67],"height:1.1569em;vertical-align:-0.25em;",[28,10645,10647,10650],{"className":10646},[5057],[28,10648,10614],{"className":10649},[5057],[28,10651,10653],{"className":10652},[508],[28,10654,10656],{"className":10655},[512],[28,10657,10659],{"className":10658},[517],[28,10660,10663],{"className":10661,"style":10662},[521],"height:0.9069em;",[28,10664,10666,10669],{"style":10665},"top:-3.1208em;margin-right:0.05em;",[28,10667],{"className":10668,"style":530},[529],[28,10670,10672],{"className":10671},[534,535,536,537],[28,10673,10675,10678,10681],{"className":10674},[72,537],[28,10676,469],{"className":10677},[72,503,537],[28,10679,93],{"className":10680},[814,537],[28,10682,619],{"className":10683},[72,537],[28,10685,370],{"className":10686},[120],[28,10688,9498],{"className":10689,"style":706},[72,503],[28,10691,318],{"className":10692},[142],[28,10694,10696],{"className":10695,"translate":32},[1034],[28,10697,10699,10774],{"className":10698,"translate":32},[31],[28,10700,10702],{"className":10701},[36],[38,10703,10704],{"xmlns":40,"display":1043},[42,10705,10706,10771],{},[45,10707,10708,10718,10720,10722,10724,10726,10728,10730,10733,10742,10758,10760,10763,10765,10767,10769],{},[45,10709,10710,10712,10715],{},[458,10711,2075],{"mathvariant":482},[458,10713,10714],{"mathvariant":482},"o",[458,10716,10717],{"mathvariant":482},"l",[87,10719,370],{"stretchy":89},[458,10721,10371],{},[87,10723,370],{"stretchy":89},[458,10725,9498],{},[87,10727,318],{"stretchy":89},[87,10729,318],{"stretchy":89},[87,10731,10732],{},"∝",[455,10734,10735,10738,10740],{},[87,10736,10737],{},"∫",[48,10739,1061],{},[458,10741,9498],{},[656,10743,10744,10750],{},[45,10745,10746,10748],{},[458,10747,10614],{},[87,10749,5000],{},[45,10751,10752,10754,10756],{},[458,10753,469],{},[87,10755,93],{},[48,10757,619],{},[87,10759,370],{"stretchy":89},[458,10761,10762],{},"ϕ",[87,10764,318],{"stretchy":89},[87,10766,98],{"separator":59},[458,10768,469],{},[458,10770,10762],{},[52,10772,10773],{"encoding":54},"\\mathrm{Vol}(B(\\theta)) \\propto \\int_0^\\theta \\sin^{d-2}(\\phi), d\\phi",[28,10775,10777,10816],{"className":10776,"ariaHidden":59},[58],[28,10778,10780,10783,10791,10794,10797,10800,10803,10807,10810,10813],{"className":10779},[63],[28,10781],{"className":10782,"style":116},[67],[28,10784,10786],{"className":10785},[72],[28,10787,10790],{"className":10788},[72,10789],"mathrm","Vol",[28,10792,370],{"className":10793},[120],[28,10795,10371],{"className":10796,"style":10436},[72,503],[28,10798,370],{"className":10799},[120],[28,10801,9498],{"className":10802,"style":706},[72,503],[28,10804,10806],{"className":10805},[142],"))",[28,10808],{"className":10809,"style":584},[134],[28,10811,10732],{"className":10812},[563],[28,10814],{"className":10815,"style":584},[134],[28,10817,10819,10823,10879,10882,10920,10923,10926,10929,10932,10935,10938],{"className":10818},[63],[28,10820],{"className":10821,"style":10822},[67],"height:2.511em;vertical-align:-0.9119em;",[28,10824,10826,10830],{"className":10825},[5057],[28,10827,10737],{"className":10828,"style":10829},[5057,5715,5716],"margin-right:0.44445em;position:relative;top:-0.0011em;",[28,10831,10833],{"className":10832},[508],[28,10834,10836,10870],{"className":10835},[512,513],[28,10837,10839,10867],{"className":10838},[517],[28,10840,10843,10855],{"className":10841,"style":10842},[521],"height:1.599em;",[28,10844,10846,10849],{"style":10845},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[28,10847],{"className":10848,"style":530},[529],[28,10850,10852],{"className":10851},[534,535,536,537],[28,10853,1061],{"className":10854},[72,537],[28,10856,10858,10861],{"style":10857},"top:-3.8129em;margin-right:0.05em;",[28,10859],{"className":10860,"style":530},[529],[28,10862,10864],{"className":10863},[534,535,536,537],[28,10865,9498],{"className":10866,"style":706},[72,503,537],[28,10868,571],{"className":10869},[570],[28,10871,10873],{"className":10872},[517],[28,10874,10877],{"className":10875,"style":10876},[521],"height:0.9119em;",[28,10878],{},[28,10880],{"className":10881,"style":135},[134],[28,10883,10885,10888],{"className":10884},[5057],[28,10886,10614],{"className":10887},[5057],[28,10889,10891],{"className":10890},[508],[28,10892,10894],{"className":10893},[512],[28,10895,10897],{"className":10896},[517],[28,10898,10900],{"className":10899,"style":10662},[521],[28,10901,10902,10905],{"style":10665},[28,10903],{"className":10904,"style":530},[529],[28,10906,10908],{"className":10907},[534,535,536,537],[28,10909,10911,10914,10917],{"className":10910},[72,537],[28,10912,469],{"className":10913},[72,503,537],[28,10915,93],{"className":10916},[814,537],[28,10918,619],{"className":10919},[72,537],[28,10921,370],{"className":10922},[120],[28,10924,10762],{"className":10925},[72,503],[28,10927,318],{"className":10928},[142],[28,10930,98],{"className":10931},[130],[28,10933],{"className":10934,"style":135},[134],[28,10936,469],{"className":10937},[72,503],[28,10939,10762],{"className":10940},[72,503],[10,10942,10943],{},"Now observe the key phenomenon:",[3994,10945,10946,11070],{},[3997,10947,4046,10948,10976,10977,11069],{},[28,10949,10951,10964],{"className":10950,"translate":32},[31],[28,10952,10954],{"className":10953},[36],[38,10955,10956],{"xmlns":40},[42,10957,10958,10962],{},[45,10959,10960],{},[458,10961,469],{},[52,10963,469],{"encoding":54},[28,10965,10967],{"className":10966,"ariaHidden":59},[58],[28,10968,10970,10973],{"className":10969},[63],[28,10971],{"className":10972,"style":3585},[67],[28,10974,469],{"className":10975},[72,503]," is small, ",[28,10978,10980,11013],{"className":10979,"translate":32},[31],[28,10981,10983],{"className":10982},[36],[38,10984,10985],{"xmlns":40},[42,10986,10987,11011],{},[45,10988,10989,11005,11007,11009],{},[656,10990,10991,10997],{},[45,10992,10993,10995],{},[458,10994,10614],{},[87,10996,5000],{},[45,10998,10999,11001,11003],{},[458,11000,469],{},[87,11002,93],{},[48,11004,619],{},[87,11006,370],{"stretchy":89},[458,11008,9498],{},[87,11010,318],{"stretchy":89},[52,11012,10633],{"encoding":54},[28,11014,11016],{"className":11015,"ariaHidden":59},[58],[28,11017,11019,11022,11060,11063,11066],{"className":11018},[63],[28,11020],{"className":11021,"style":10643},[67],[28,11023,11025,11028],{"className":11024},[5057],[28,11026,10614],{"className":11027},[5057],[28,11029,11031],{"className":11030},[508],[28,11032,11034],{"className":11033},[512],[28,11035,11037],{"className":11036},[517],[28,11038,11040],{"className":11039,"style":10662},[521],[28,11041,11042,11045],{"style":10665},[28,11043],{"className":11044,"style":530},[529],[28,11046,11048],{"className":11047},[534,535,536,537],[28,11049,11051,11054,11057],{"className":11050},[72,537],[28,11052,469],{"className":11053},[72,503,537],[28,11055,93],{"className":11056},[814,537],[28,11058,619],{"className":11059},[72,537],[28,11061,370],{"className":11062},[120],[28,11064,9498],{"className":11065,"style":706},[72,503],[28,11067,318],{"className":11068},[142]," is relatively flat",[3997,11071,4046,11072,11123,11124,11216,11217],{},[28,11073,11075,11093],{"className":11074,"translate":32},[31],[28,11076,11078],{"className":11077},[36],[38,11079,11080],{"xmlns":40},[42,11081,11082,11090],{},[45,11083,11084,11086,11088],{},[458,11085,469],{},[87,11087,3459],{},[458,11089,4936],{"mathvariant":482},[52,11091,11092],{"encoding":54},"d \\to \\infty",[28,11094,11096,11114],{"className":11095,"ariaHidden":59},[58],[28,11097,11099,11102,11105,11108,11111],{"className":11098},[63],[28,11100],{"className":11101,"style":3585},[67],[28,11103,469],{"className":11104},[72,503],[28,11106],{"className":11107,"style":584},[134],[28,11109,3459],{"className":11110},[563],[28,11112],{"className":11113,"style":584},[134],[28,11115,11117,11120],{"className":11116},[63],[28,11118],{"className":11119,"style":3615},[67],[28,11121,4936],{"className":11122},[72],", ",[28,11125,11127,11160],{"className":11126,"translate":32},[31],[28,11128,11130],{"className":11129},[36],[38,11131,11132],{"xmlns":40},[42,11133,11134,11158],{},[45,11135,11136,11152,11154,11156],{},[656,11137,11138,11144],{},[45,11139,11140,11142],{},[458,11141,10614],{},[87,11143,5000],{},[45,11145,11146,11148,11150],{},[458,11147,469],{},[87,11149,93],{},[48,11151,619],{},[87,11153,370],{"stretchy":89},[458,11155,9498],{},[87,11157,318],{"stretchy":89},[52,11159,10633],{"encoding":54},[28,11161,11163],{"className":11162,"ariaHidden":59},[58],[28,11164,11166,11169,11207,11210,11213],{"className":11165},[63],[28,11167],{"className":11168,"style":10643},[67],[28,11170,11172,11175],{"className":11171},[5057],[28,11173,10614],{"className":11174},[5057],[28,11176,11178],{"className":11177},[508],[28,11179,11181],{"className":11180},[512],[28,11182,11184],{"className":11183},[517],[28,11185,11187],{"className":11186,"style":10662},[521],[28,11188,11189,11192],{"style":10665},[28,11190],{"className":11191,"style":530},[529],[28,11193,11195],{"className":11194},[534,535,536,537],[28,11196,11198,11201,11204],{"className":11197},[72,537],[28,11199,469],{"className":11200},[72,503,537],[28,11202,93],{"className":11203},[814,537],[28,11205,619],{"className":11206},[72,537],[28,11208,370],{"className":11209},[120],[28,11211,9498],{"className":11212,"style":706},[72,503],[28,11214,318],{"className":11215},[142]," becomes sharply peaked at:\n",[28,11218,11220],{"className":11219,"translate":32},[1034],[28,11221,11223,11245],{"className":11222,"translate":32},[31],[28,11224,11226],{"className":11225},[36],[38,11227,11228],{"xmlns":40,"display":1043},[42,11229,11230,11242],{},[45,11231,11232,11234,11236],{},[458,11233,9498],{},[87,11235,472],{},[1187,11237,11238,11240],{},[458,11239,654],{},[48,11241,619],{},[52,11243,11244],{"encoding":54},"\\theta = \\frac{\\pi}{2}",[28,11246,11248,11266],{"className":11247,"ariaHidden":59},[58],[28,11249,11251,11254,11257,11260,11263],{"className":11250},[63],[28,11252],{"className":11253,"style":3585},[67],[28,11255,9498],{"className":11256,"style":706},[72,503],[28,11258],{"className":11259,"style":584},[134],[28,11261,472],{"className":11262},[563],[28,11264],{"className":11265,"style":584},[134],[28,11267,11269,11273],{"className":11268},[63],[28,11270],{"className":11271,"style":11272},[67],"height:1.7936em;vertical-align:-0.686em;",[28,11274,11276,11279,11333],{"className":11275},[72],[28,11277],{"className":11278},[120,1753],[28,11280,11282],{"className":11281},[1187],[28,11283,11285,11325],{"className":11284},[512,513],[28,11286,11288,11322],{"className":11287},[517],[28,11289,11292,11303,11311],{"className":11290,"style":11291},[521],"height:1.1076em;",[28,11293,11294,11297],{"style":1769},[28,11295],{"className":11296,"style":1773},[529],[28,11298,11300],{"className":11299},[72],[28,11301,619],{"className":11302},[72],[28,11304,11305,11308],{"style":1822},[28,11306],{"className":11307,"style":1773},[529],[28,11309],{"className":11310,"style":1830},[1829],[28,11312,11313,11316],{"style":1833},[28,11314],{"className":11315,"style":1773},[529],[28,11317,11319],{"className":11318},[72],[28,11320,654],{"className":11321,"style":699},[72,503],[28,11323,571],{"className":11324},[570],[28,11326,11328],{"className":11327},[517],[28,11329,11331],{"className":11330,"style":1968},[521],[28,11332],{},[28,11334],{"className":11335},[142,1753],[10,11337,11338],{},"This leads to: Most of the mass concentrates near the equator orthogonal to any fixed direction.",[10,11340,11341,11342,10048],{},"More formally, for any fixed ",[28,11343,11345,11365],{"className":11344,"translate":32},[31],[28,11346,11348],{"className":11347},[36],[38,11349,11350],{"xmlns":40},[42,11351,11352,11362],{},[45,11353,11354,11357,11360],{},[458,11355,11356],{},"ϵ",[87,11358,11359],{},">",[48,11361,1061],{},[52,11363,11364],{"encoding":54},"\\epsilon > 0",[28,11366,11368,11387],{"className":11367,"ariaHidden":59},[58],[28,11369,11371,11375,11378,11381,11384],{"className":11370},[63],[28,11372],{"className":11373,"style":11374},[67],"height:0.5782em;vertical-align:-0.0391em;",[28,11376,11356],{"className":11377},[72,503],[28,11379],{"className":11380,"style":584},[134],[28,11382,11359],{"className":11383},[563],[28,11385],{"className":11386,"style":584},[134],[28,11388,11390,11393],{"className":11389},[63],[28,11391],{"className":11392,"style":68},[67],[28,11394,1061],{"className":11395},[72],[28,11397,11399],{"className":11398,"translate":32},[1034],[28,11400,11402,11471],{"className":11401,"translate":32},[31],[28,11403,11405],{"className":11404},[36],[38,11406,11407],{"xmlns":40,"display":1043},[42,11408,11409,11468],{},[45,11410,11411,11413,11453,11455,11457,11459,11462,11464,11466],{},[458,11412,460],{"mathvariant":5307},[45,11414,11415,11417,11447,11449,11451],{},[87,11416,370],{"fence":59},[45,11418,11419,11421,11423,11425,11427,11429,11435,11437,11439,11445],{},[87,11420,5561],{"fence":59},[458,11422,10402],{"mathvariant":482},[87,11424,370],{"stretchy":89},[458,11426,747],{},[87,11428,98],{"separator":59},[1065,11430,11431,11433],{},[458,11432,747],{},[48,11434,1061],{},[87,11436,318],{"stretchy":89},[87,11438,93],{},[1187,11440,11441,11443],{},[458,11442,654],{},[48,11444,619],{},[87,11446,5561],{"fence":59},[87,11448,762],{},[458,11450,11356],{},[87,11452,318],{"fence":59},[87,11454,3459],{},[48,11456,50],{},[134,11458],{"width":6974},[462,11460,11461],{},"as ",[458,11463,469],{},[87,11465,3459],{},[458,11467,4936],{"mathvariant":482},[52,11469,11470],{"encoding":54},"\\mathbb{P}\\left( \\left| \\angle(x, x_0) - \\frac{\\pi}{2} \\right| \\leq \\epsilon \\right)\n\\to 1 \\quad \\text{as } d \\to \\infty",[28,11472,11474,11740,11770],{"className":11473,"ariaHidden":59},[58],[28,11475,11477,11481,11484,11487,11731,11734,11737],{"className":11476},[63],[28,11478],{"className":11479,"style":11480},[67],"height:1.836em;vertical-align:-0.686em;",[28,11482,460],{"className":11483},[72,5329],[28,11485],{"className":11486,"style":135},[134],[28,11488,11490,11497,11713,11716,11719,11722,11725],{"className":11489},[3795],[28,11491,11493],{"className":11492,"style":3800},[120,3799],[28,11494,370],{"className":11495},[3804,11496],"size2",[28,11498,11500,11547,11550,11553,11556,11559,11562,11602,11605,11608,11611,11614,11676],{"className":11499},[3795],[28,11501,11503],{"className":11502},[120],[28,11504,11507],{"className":11505},[3804,11506],"mult",[28,11508,11510,11538],{"className":11509},[512,513],[28,11511,11513,11535],{"className":11512},[517],[28,11514,11517],{"className":11515,"style":11516},[521],"height:1.15em;",[28,11518,11520,11524],{"style":11519},"top:-3.15em;",[28,11521],{"className":11522,"style":11523},[529],"height:3.8em;",[28,11525,11527],{"style":11526},"width:0.333em;height:1.800em;",[4471,11528,11532],{"xmlns":4473,"width":11529,"height":11530,"viewBox":11531},"0.333em","1.800em","0 0 333 1800",[4479,11533],{"d":11534},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[28,11536,571],{"className":11537},[570],[28,11539,11541],{"className":11540},[517],[28,11542,11545],{"className":11543,"style":11544},[521],"height:0.65em;",[28,11546],{},[28,11548,10402],{"className":11549},[72],[28,11551,370],{"className":11552},[120],[28,11554,747],{"className":11555},[72,503],[28,11557,98],{"className":11558},[130],[28,11560],{"className":11561,"style":135},[134],[28,11563,11565,11568],{"className":11564},[72],[28,11566,747],{"className":11567},[72,503],[28,11569,11571],{"className":11570},[508],[28,11572,11574,11594],{"className":11573},[512,513],[28,11575,11577,11591],{"className":11576},[517],[28,11578,11580],{"className":11579,"style":9319},[521],[28,11581,11582,11585],{"style":7358},[28,11583],{"className":11584,"style":530},[529],[28,11586,11588],{"className":11587},[534,535,536,537],[28,11589,1061],{"className":11590},[72,537],[28,11592,571],{"className":11593},[570],[28,11595,11597],{"className":11596},[517],[28,11598,11600],{"className":11599,"style":1306},[521],[28,11601],{},[28,11603,318],{"className":11604},[142],[28,11606],{"className":11607,"style":810},[134],[28,11609,93],{"className":11610},[814],[28,11612],{"className":11613,"style":810},[134],[28,11615,11617,11620,11673],{"className":11616},[72],[28,11618],{"className":11619},[120,1753],[28,11621,11623],{"className":11622},[1187],[28,11624,11626,11665],{"className":11625},[512,513],[28,11627,11629,11662],{"className":11628},[517],[28,11630,11632,11643,11651],{"className":11631,"style":11291},[521],[28,11633,11634,11637],{"style":1769},[28,11635],{"className":11636,"style":1773},[529],[28,11638,11640],{"className":11639},[72],[28,11641,619],{"className":11642},[72],[28,11644,11645,11648],{"style":1822},[28,11646],{"className":11647,"style":1773},[529],[28,11649],{"className":11650,"style":1830},[1829],[28,11652,11653,11656],{"style":1833},[28,11654],{"className":11655,"style":1773},[529],[28,11657,11659],{"className":11658},[72],[28,11660,654],{"className":11661,"style":699},[72,503],[28,11663,571],{"className":11664},[570],[28,11666,11668],{"className":11667},[517],[28,11669,11671],{"className":11670,"style":1968},[521],[28,11672],{},[28,11674],{"className":11675},[142,1753],[28,11677,11679],{"className":11678},[142],[28,11680,11682],{"className":11681},[3804,11506],[28,11683,11685,11705],{"className":11684},[512,513],[28,11686,11688,11702],{"className":11687},[517],[28,11689,11691],{"className":11690,"style":11516},[521],[28,11692,11693,11696],{"style":11519},[28,11694],{"className":11695,"style":11523},[529],[28,11697,11698],{"style":11526},[4471,11699,11700],{"xmlns":4473,"width":11529,"height":11530,"viewBox":11531},[4479,11701],{"d":11534},[28,11703,571],{"className":11704},[570],[28,11706,11708],{"className":11707},[517],[28,11709,11711],{"className":11710,"style":11544},[521],[28,11712],{},[28,11714],{"className":11715,"style":584},[134],[28,11717,762],{"className":11718},[563],[28,11720],{"className":11721,"style":584},[134],[28,11723,11356],{"className":11724},[72,503],[28,11726,11728],{"className":11727,"style":3800},[142,3799],[28,11729,318],{"className":11730},[3804,11496],[28,11732],{"className":11733,"style":584},[134],[28,11735,3459],{"className":11736},[563],[28,11738],{"className":11739,"style":584},[134],[28,11741,11743,11746,11749,11752,11758,11761,11764,11767],{"className":11742},[63],[28,11744],{"className":11745,"style":3585},[67],[28,11747,50],{"className":11748},[72],[28,11750],{"className":11751,"style":7067},[134],[28,11753,11755],{"className":11754},[72,541],[28,11756,11461],{"className":11757},[72],[28,11759,469],{"className":11760},[72,503],[28,11762],{"className":11763,"style":584},[134],[28,11765,3459],{"className":11766},[563],[28,11768],{"className":11769,"style":584},[134],[28,11771,11773,11776],{"className":11772},[63],[28,11774],{"className":11775,"style":3615},[67],[28,11777,4936],{"className":11778},[72],[10,11780,11781],{},"This is a manifestation of the concentration of measure phenomenon on positively curved spaces.",[10,11783,11784],{},"The Geometric Interpretation could be described as “Equatorial Collapse”. In Euclidean space, samples concentrate in a thin outer shell. On the sphere, samples instead concentrate in a thin equatorial band.",[10,11786,11787],{},"This can be viewed as a curvature-induced redistribution of measure.",[5284,11789,11791],{"id":11790},"angular-distance-in-positive-curvature","Angular Distance in Positive Curvature",[10,11793,11794],{},"Let:",[28,11796,11798],{"className":11797,"translate":32},[1034],[28,11799,11801,11840],{"className":11800,"translate":32},[31],[28,11802,11804],{"className":11803},[36],[38,11805,11806],{"xmlns":40,"display":1043},[42,11807,11808,11837],{},[45,11809,11810,11812,11814,11816,11818,11821,11823,11835],{},[458,11811,747],{},[87,11813,98],{"separator":59},[458,11815,757],{},[87,11817,5377],{},[462,11819,11820],{},"Uniform",[87,11822,370],{"stretchy":89},[656,11824,11825,11827],{},[458,11826,649],{"mathvariant":5307},[45,11828,11829,11831,11833],{},[458,11830,469],{},[87,11832,93],{},[48,11834,50],{},[87,11836,318],{"stretchy":89},[52,11838,11839],{"encoding":54},"x, y \\sim \\text{Uniform}(\\mathbb{S}^{d-1})",[28,11841,11843,11870],{"className":11842,"ariaHidden":59},[58],[28,11844,11846,11849,11852,11855,11858,11861,11864,11867],{"className":11845},[63],[28,11847],{"className":11848,"style":5412},[67],[28,11850,747],{"className":11851},[72,503],[28,11853,98],{"className":11854},[130],[28,11856],{"className":11857,"style":135},[134],[28,11859,757],{"className":11860,"style":699},[72,503],[28,11862],{"className":11863,"style":584},[134],[28,11865,5377],{"className":11866},[563],[28,11868],{"className":11869,"style":584},[134],[28,11871,11873,11876,11882,11885,11923],{"className":11872},[63],[28,11874],{"className":11875,"style":3703},[67],[28,11877,11879],{"className":11878},[72,541],[28,11880,11820],{"className":11881},[72],[28,11883,370],{"className":11884},[120],[28,11886,11888,11891],{"className":11887},[72],[28,11889,649],{"className":11890},[72,5329],[28,11892,11894],{"className":11893},[508],[28,11895,11897],{"className":11896},[512],[28,11898,11900],{"className":11899},[517],[28,11901,11903],{"className":11902,"style":1690},[521],[28,11904,11905,11908],{"style":1357},[28,11906],{"className":11907,"style":530},[529],[28,11909,11911],{"className":11910},[534,535,536,537],[28,11912,11914,11917,11920],{"className":11913},[72,537],[28,11915,469],{"className":11916},[72,503,537],[28,11918,93],{"className":11919},[814,537],[28,11921,50],{"className":11922},[72,537],[28,11924,318],{"className":11925},[142],[10,11927,11928],{},"Then their inner product satisfies:",[28,11930,11932],{"className":11931,"translate":32},[1034],[28,11933,11935,11977],{"className":11934,"translate":32},[31],[28,11936,11938],{"className":11937},[36],[38,11939,11940],{"xmlns":40,"display":1043},[42,11941,11942,11974],{},[45,11943,11944,11946,11948,11950,11952,11954,11956,11958],{},[87,11945,6714],{"stretchy":89},[458,11947,747],{},[87,11949,98],{"separator":59},[458,11951,757],{},[87,11953,6723],{"stretchy":89},[87,11955,5377],{},[458,11957,5384],{"mathvariant":5383},[45,11959,11960,11962,11964,11966,11972],{},[87,11961,370],{"fence":59},[48,11963,1061],{},[87,11965,98],{"separator":59},[1187,11967,11968,11970],{},[48,11969,50],{},[458,11971,469],{},[87,11973,318],{"fence":59},[52,11975,11976],{"encoding":54},"\\langle x, y \\rangle \\sim \\mathcal{N}\\left(0, \\frac{1}{d}\\right)",[28,11978,11980,12013],{"className":11979,"ariaHidden":59},[58],[28,11981,11983,11986,11989,11992,11995,11998,12001,12004,12007,12010],{"className":11982},[63],[28,11984],{"className":11985,"style":116},[67],[28,11987,6714],{"className":11988},[120],[28,11990,747],{"className":11991},[72,503],[28,11993,98],{"className":11994},[130],[28,11996],{"className":11997,"style":135},[134],[28,11999,757],{"className":12000,"style":699},[72,503],[28,12002,6723],{"className":12003},[142],[28,12005],{"className":12006,"style":584},[134],[28,12008,5377],{"className":12009},[563],[28,12011],{"className":12012,"style":584},[134],[28,12014,12016,12020,12023,12026],{"className":12015},[63],[28,12017],{"className":12018,"style":12019},[67],"height:2.4em;vertical-align:-0.95em;",[28,12021,5384],{"className":12022,"style":5455},[72,5454],[28,12024],{"className":12025,"style":135},[134],[28,12027,12029,12035,12038,12041,12044,12106],{"className":12028},[3795],[28,12030,12032],{"className":12031,"style":3800},[120,3799],[28,12033,370],{"className":12034},[3804,536],[28,12036,1061],{"className":12037},[72],[28,12039,98],{"className":12040},[130],[28,12042],{"className":12043,"style":135},[134],[28,12045,12047,12050,12103],{"className":12046},[72],[28,12048],{"className":12049},[120,1753],[28,12051,12053],{"className":12052},[1187],[28,12054,12056,12095],{"className":12055},[512,513],[28,12057,12059,12092],{"className":12058},[517],[28,12060,12062,12073,12081],{"className":12061,"style":1925},[521],[28,12063,12064,12067],{"style":1769},[28,12065],{"className":12066,"style":1773},[529],[28,12068,12070],{"className":12069},[72],[28,12071,469],{"className":12072},[72,503],[28,12074,12075,12078],{"style":1822},[28,12076],{"className":12077,"style":1773},[529],[28,12079],{"className":12080,"style":1830},[1829],[28,12082,12083,12086],{"style":1833},[28,12084],{"className":12085,"style":1773},[529],[28,12087,12089],{"className":12088},[72],[28,12090,50],{"className":12091},[72],[28,12093,571],{"className":12094},[570],[28,12096,12098],{"className":12097},[517],[28,12099,12101],{"className":12100,"style":1968},[521],[28,12102],{},[28,12104],{"className":12105},[142,1753],[28,12107,12109],{"className":12108,"style":3800},[142,3799],[28,12110,318],{"className":12111},[3804,536],[10,12113,8553],{},[28,12115,12117],{"className":12116,"translate":32},[1034],[28,12118,12120,12171],{"className":12119,"translate":32},[31],[28,12121,12123],{"className":12122},[36],[38,12124,12125],{"xmlns":40,"display":1043},[42,12126,12127,12168],{},[45,12128,12129,12131,12133,12135,12137,12139,12141,12143,12145,12147,12149,12151,12153,12156,12158,12160,12162],{},[458,12130,9493],{},[87,12132,5000],{},[458,12134,9498],{},[87,12136,472],{},[87,12138,6714],{"stretchy":89},[458,12140,747],{},[87,12142,98],{"separator":59},[458,12144,757],{},[87,12146,6723],{"stretchy":89},[87,12148,6548],{},[48,12150,1061],{},[134,12152],{"width":6974},[87,12154,12155],{},"⇒",[134,12157],{"width":6974},[458,12159,9498],{},[87,12161,6548],{},[1187,12163,12164,12166],{},[458,12165,654],{},[48,12167,619],{},[52,12169,12170],{"encoding":54},"\\cos\\theta = \\langle x, y \\rangle \\approx 0\n\\quad \\Rightarrow \\quad\n\\theta \\approx \\frac{\\pi}{2}",[28,12172,12174,12198,12231,12255,12273],{"className":12173,"ariaHidden":59},[58],[28,12175,12177,12180,12183,12186,12189,12192,12195],{"className":12176},[63],[28,12178],{"className":12179,"style":3585},[67],[28,12181,9493],{"className":12182},[5057],[28,12184],{"className":12185,"style":135},[134],[28,12187,9498],{"className":12188,"style":706},[72,503],[28,12190],{"className":12191,"style":584},[134],[28,12193,472],{"className":12194},[563],[28,12196],{"className":12197,"style":584},[134],[28,12199,12201,12204,12207,12210,12213,12216,12219,12222,12225,12228],{"className":12200},[63],[28,12202],{"className":12203,"style":116},[67],[28,12205,6714],{"className":12206},[120],[28,12208,747],{"className":12209},[72,503],[28,12211,98],{"className":12212},[130],[28,12214],{"className":12215,"style":135},[134],[28,12217,757],{"className":12218,"style":699},[72,503],[28,12220,6723],{"className":12221},[142],[28,12223],{"className":12224,"style":584},[134],[28,12226,6548],{"className":12227},[563],[28,12229],{"className":12230,"style":584},[134],[28,12232,12234,12237,12240,12243,12246,12249,12252],{"className":12233},[63],[28,12235],{"className":12236,"style":68},[67],[28,12238,1061],{"className":12239},[72],[28,12241],{"className":12242,"style":7067},[134],[28,12244],{"className":12245,"style":584},[134],[28,12247,12155],{"className":12248},[563],[28,12250],{"className":12251,"style":7067},[134],[28,12253],{"className":12254,"style":584},[134],[28,12256,12258,12261,12264,12267,12270],{"className":12257},[63],[28,12259],{"className":12260,"style":3585},[67],[28,12262,9498],{"className":12263,"style":706},[72,503],[28,12265],{"className":12266,"style":584},[134],[28,12268,6548],{"className":12269},[563],[28,12271],{"className":12272,"style":584},[134],[28,12274,12276,12279],{"className":12275},[63],[28,12277],{"className":12278,"style":11272},[67],[28,12280,12282,12285,12338],{"className":12281},[72],[28,12283],{"className":12284},[120,1753],[28,12286,12288],{"className":12287},[1187],[28,12289,12291,12330],{"className":12290},[512,513],[28,12292,12294,12327],{"className":12293},[517],[28,12295,12297,12308,12316],{"className":12296,"style":11291},[521],[28,12298,12299,12302],{"style":1769},[28,12300],{"className":12301,"style":1773},[529],[28,12303,12305],{"className":12304},[72],[28,12306,619],{"className":12307},[72],[28,12309,12310,12313],{"style":1822},[28,12311],{"className":12312,"style":1773},[529],[28,12314],{"className":12315,"style":1830},[1829],[28,12317,12318,12321],{"style":1833},[28,12319],{"className":12320,"style":1773},[529],[28,12322,12324],{"className":12323},[72],[28,12325,654],{"className":12326,"style":699},[72,503],[28,12328,571],{"className":12329},[570],[28,12331,12333],{"className":12332},[517],[28,12334,12336],{"className":12335,"style":1968},[521],[28,12337],{},[28,12339],{"className":12340},[142,1753],[10,12342,12343],{},"So we have the conclusion:",[12345,12346,12347],"blockquote",{},[10,12348,12349],{},"Two random points on a high-dimensional sphere are almost always orthogonal.",[10,12351,12352],{},"This implies:",[28,12354,12356],{"className":12355,"translate":32},[1034],[28,12357,12359,12417],{"className":12358,"translate":32},[31],[28,12360,12362],{"className":12361},[36],[38,12363,12364],{"xmlns":40,"display":1043},[42,12365,12366,12414],{},[45,12367,12368,12375,12377,12379,12381,12383,12385,12387,12390,12392,12394,12396,12398,12400,12402,12404,12406,12408],{},[1065,12369,12370,12372],{},[458,12371,469],{},[462,12373,12374],{},"geo",[87,12376,370],{"stretchy":89},[458,12378,747],{},[87,12380,98],{"separator":59},[458,12382,757],{},[87,12384,318],{"stretchy":89},[87,12386,472],{},[458,12388,12389],{},"arccos",[87,12391,5000],{},[87,12393,370],{"stretchy":89},[87,12395,6714],{"stretchy":89},[458,12397,747],{},[87,12399,98],{"separator":59},[458,12401,757],{},[87,12403,6723],{"stretchy":89},[87,12405,318],{"stretchy":89},[87,12407,6548],{},[1187,12409,12410,12412],{},[458,12411,654],{},[48,12413,619],{},[52,12415,12416],{"encoding":54},"d_{\\text{geo}}(x, y) = \\arccos(\\langle x, y \\rangle) \\approx \\frac{\\pi}{2}",[28,12418,12420,12502,12540],{"className":12419,"ariaHidden":59},[58],[28,12421,12423,12427,12475,12478,12481,12484,12487,12490,12493,12496,12499],{"className":12422},[63],[28,12424],{"className":12425,"style":12426},[67],"height:1.0361em;vertical-align:-0.2861em;",[28,12428,12430,12433],{"className":12429},[72],[28,12431,469],{"className":12432},[72,503],[28,12434,12436],{"className":12435},[508],[28,12437,12439,12466],{"className":12438},[512,513],[28,12440,12442,12463],{"className":12441},[517],[28,12443,12446],{"className":12444,"style":12445},[521],"height:0.1514em;",[28,12447,12448,12451],{"style":7358},[28,12449],{"className":12450,"style":530},[529],[28,12452,12454],{"className":12453},[534,535,536,537],[28,12455,12457],{"className":12456},[72,537],[28,12458,12460],{"className":12459},[72,541,537],[28,12461,12374],{"className":12462},[72,537],[28,12464,571],{"className":12465},[570],[28,12467,12469],{"className":12468},[517],[28,12470,12473],{"className":12471,"style":12472},[521],"height:0.2861em;",[28,12474],{},[28,12476,370],{"className":12477},[120],[28,12479,747],{"className":12480},[72,503],[28,12482,98],{"className":12483},[130],[28,12485],{"className":12486,"style":135},[134],[28,12488,757],{"className":12489,"style":699},[72,503],[28,12491,318],{"className":12492},[142],[28,12494],{"className":12495,"style":584},[134],[28,12497,472],{"className":12498},[563],[28,12500],{"className":12501,"style":584},[134],[28,12503,12505,12508,12511,12515,12518,12521,12524,12527,12531,12534,12537],{"className":12504},[63],[28,12506],{"className":12507,"style":116},[67],[28,12509,12389],{"className":12510},[5057],[28,12512,12514],{"className":12513},[120],"(⟨",[28,12516,747],{"className":12517},[72,503],[28,12519,98],{"className":12520},[130],[28,12522],{"className":12523,"style":135},[134],[28,12525,757],{"className":12526,"style":699},[72,503],[28,12528,12530],{"className":12529},[142],"⟩)",[28,12532],{"className":12533,"style":584},[134],[28,12535,6548],{"className":12536},[563],[28,12538],{"className":12539,"style":584},[134],[28,12541,12543,12546],{"className":12542},[63],[28,12544],{"className":12545,"style":11272},[67],[28,12547,12549,12552,12605],{"className":12548},[72],[28,12550],{"className":12551},[120,1753],[28,12553,12555],{"className":12554},[1187],[28,12556,12558,12597],{"className":12557},[512,513],[28,12559,12561,12594],{"className":12560},[517],[28,12562,12564,12575,12583],{"className":12563,"style":11291},[521],[28,12565,12566,12569],{"style":1769},[28,12567],{"className":12568,"style":1773},[529],[28,12570,12572],{"className":12571},[72],[28,12573,619],{"className":12574},[72],[28,12576,12577,12580],{"style":1822},[28,12578],{"className":12579,"style":1773},[529],[28,12581],{"className":12582,"style":1830},[1829],[28,12584,12585,12588],{"style":1833},[28,12586],{"className":12587,"style":1773},[529],[28,12589,12591],{"className":12590},[72],[28,12592,654],{"className":12593,"style":699},[72,503],[28,12595,571],{"className":12596},[570],[28,12598,12600],{"className":12599},[517],[28,12601,12603],{"className":12602,"style":1968},[521],[28,12604],{},[28,12606],{"className":12607},[142,1753],[5284,12609,12611],{"id":12610},"distance-concentration-under-geodesic-metric","Distance Concentration under Geodesic Metric",[10,12613,12614,12615,12689],{},"Unlike Euclidean distance, the natural metric on ",[28,12616,12618,12642],{"className":12617,"translate":32},[31],[28,12619,12621],{"className":12620},[36],[38,12622,12623],{"xmlns":40},[42,12624,12625,12639],{},[45,12626,12627],{},[656,12628,12629,12631],{},[458,12630,649],{"mathvariant":5307},[45,12632,12633,12635,12637],{},[458,12634,469],{},[87,12636,93],{},[48,12638,50],{},[52,12640,12641],{"encoding":54},"\\mathbb{S}^{d-1}",[28,12643,12645],{"className":12644,"ariaHidden":59},[58],[28,12646,12648,12651],{"className":12647},[63],[28,12649],{"className":12650,"style":522},[67],[28,12652,12654,12657],{"className":12653},[72],[28,12655,649],{"className":12656},[72,5329],[28,12658,12660],{"className":12659},[508],[28,12661,12663],{"className":12662},[512],[28,12664,12666],{"className":12665},[517],[28,12667,12669],{"className":12668,"style":522},[521],[28,12670,12671,12674],{"style":547},[28,12672],{"className":12673,"style":530},[529],[28,12675,12677],{"className":12676},[534,535,536,537],[28,12678,12680,12683,12686],{"className":12679},[72,537],[28,12681,469],{"className":12682},[72,503,537],[28,12684,93],{"className":12685},[814,537],[28,12687,50],{"className":12688},[72,537]," is the geodesic distance:",[28,12691,12693],{"className":12692,"translate":32},[1034],[28,12694,12696,12744],{"className":12695,"translate":32},[31],[28,12697,12699],{"className":12698},[36],[38,12700,12701],{"xmlns":40,"display":1043},[42,12702,12703,12741],{},[45,12704,12705,12711,12713,12715,12717,12719,12721,12723,12725,12727,12729,12731,12733,12735,12737,12739],{},[1065,12706,12707,12709],{},[458,12708,469],{},[458,12710,649],{"mathvariant":5307},[87,12712,370],{"stretchy":89},[458,12714,747],{},[87,12716,98],{"separator":59},[458,12718,757],{},[87,12720,318],{"stretchy":89},[87,12722,472],{},[458,12724,12389],{},[87,12726,5000],{},[87,12728,370],{"stretchy":89},[87,12730,6714],{"stretchy":89},[458,12732,747],{},[87,12734,98],{"separator":59},[458,12736,757],{},[87,12738,6723],{"stretchy":89},[87,12740,318],{"stretchy":89},[52,12742,12743],{"encoding":54},"d_{\\mathbb{S}}(x, y) = \\arccos(\\langle x, y \\rangle)",[28,12745,12747,12824],{"className":12746,"ariaHidden":59},[58],[28,12748,12750,12753,12797,12800,12803,12806,12809,12812,12815,12818,12821],{"className":12749},[63],[28,12751],{"className":12752,"style":116},[67],[28,12754,12756,12759],{"className":12755},[72],[28,12757,469],{"className":12758},[72,503],[28,12760,12762],{"className":12761},[508],[28,12763,12765,12789],{"className":12764},[512,513],[28,12766,12768,12786],{"className":12767},[517],[28,12769,12772],{"className":12770,"style":12771},[521],"height:0.3322em;",[28,12773,12774,12777],{"style":7358},[28,12775],{"className":12776,"style":530},[529],[28,12778,12780],{"className":12779},[534,535,536,537],[28,12781,12783],{"className":12782},[72,537],[28,12784,649],{"className":12785},[72,5329,537],[28,12787,571],{"className":12788},[570],[28,12790,12792],{"className":12791},[517],[28,12793,12795],{"className":12794,"style":1306},[521],[28,12796],{},[28,12798,370],{"className":12799},[120],[28,12801,747],{"className":12802},[72,503],[28,12804,98],{"className":12805},[130],[28,12807],{"className":12808,"style":135},[134],[28,12810,757],{"className":12811,"style":699},[72,503],[28,12813,318],{"className":12814},[142],[28,12816],{"className":12817,"style":584},[134],[28,12819,472],{"className":12820},[563],[28,12822],{"className":12823,"style":584},[134],[28,12825,12827,12830,12833,12836,12839,12842,12845,12848],{"className":12826},[63],[28,12828],{"className":12829,"style":116},[67],[28,12831,12389],{"className":12832},[5057],[28,12834,12514],{"className":12835},[120],[28,12837,747],{"className":12838},[72,503],[28,12840,98],{"className":12841},[130],[28,12843],{"className":12844,"style":135},[134],[28,12846,757],{"className":12847,"style":699},[72,503],[28,12849,12530],{"className":12850},[142],[10,12852,7422],{},[3994,12854,12855,12931],{},[3997,12856,12857],{},[28,12858,12860,12886],{"className":12859,"translate":32},[31],[28,12861,12863],{"className":12862},[36],[38,12864,12865],{"xmlns":40},[42,12866,12867,12883],{},[45,12868,12869,12871,12873,12875,12877,12879,12881],{},[87,12870,6714],{"stretchy":89},[458,12872,747],{},[87,12874,98],{"separator":59},[458,12876,757],{},[87,12878,6723],{"stretchy":89},[87,12880,3459],{},[48,12882,1061],{},[52,12884,12885],{"encoding":54},"\\langle x, y \\rangle \\to 0",[28,12887,12889,12922],{"className":12888,"ariaHidden":59},[58],[28,12890,12892,12895,12898,12901,12904,12907,12910,12913,12916,12919],{"className":12891},[63],[28,12893],{"className":12894,"style":116},[67],[28,12896,6714],{"className":12897},[120],[28,12899,747],{"className":12900},[72,503],[28,12902,98],{"className":12903},[130],[28,12905],{"className":12906,"style":135},[134],[28,12908,757],{"className":12909,"style":699},[72,503],[28,12911,6723],{"className":12912},[142],[28,12914],{"className":12915,"style":584},[134],[28,12917,3459],{"className":12918},[563],[28,12920],{"className":12921,"style":584},[134],[28,12923,12925,12928],{"className":12924},[63],[28,12926],{"className":12927,"style":68},[67],[28,12929,1061],{"className":12930},[72],[3997,12932,12933,12934],{},"fluctuations are ",[28,12935,12937,12963],{"className":12936,"translate":32},[31],[28,12938,12940],{"className":12939},[36],[38,12941,12942],{"xmlns":40},[42,12943,12944,12960],{},[45,12945,12946,12948,12950,12952,12954,12958],{},[458,12947,7992],{},[87,12949,370],{"stretchy":89},[48,12951,50],{},[458,12953,3661],{"mathvariant":482},[4413,12955,12956],{},[458,12957,469],{},[87,12959,318],{"stretchy":89},[52,12961,12962],{"encoding":54},"O(1\u002F\\sqrt{d})",[28,12964,12966],{"className":12965,"ariaHidden":59},[58],[28,12967,12969,12972,12975,12978,12982,13029],{"className":12968},[63],[28,12970],{"className":12971,"style":8013},[67],[28,12973,7992],{"className":12974,"style":706},[72,503],[28,12976,370],{"className":12977},[120],[28,12979,12981],{"className":12980},[72],"1\u002F",[28,12983,12985],{"className":12984},[72,4433],[28,12986,12988,13021],{"className":12987},[512,513],[28,12989,12991,13018],{"className":12990},[517],[28,12992,12994,13006],{"className":12993,"style":6602},[521],[28,12995,12997,13000],{"className":12996,"style":4448},[4447],[28,12998],{"className":12999,"style":1773},[529],[28,13001,13003],{"className":13002,"style":4455},[72],[28,13004,469],{"className":13005},[72,503],[28,13007,13008,13011],{"style":6617},[28,13009],{"className":13010,"style":1773},[529],[28,13012,13014],{"className":13013,"style":4469},[4468],[4471,13015,13016],{"xmlns":4473,"width":4474,"height":4475,"viewBox":4476,"preserveAspectRatio":4477},[4479,13017],{"d":4481},[28,13019,571],{"className":13020},[570],[28,13022,13024],{"className":13023},[517],[28,13025,13027],{"className":13026,"style":6637},[521],[28,13028],{},[28,13030,318],{"className":13031},[142],[10,13033,13034],{},"we have:",[28,13036,13038],{"className":13037,"translate":32},[1034],[28,13039,13041,13079],{"className":13040,"translate":32},[31],[28,13042,13044],{"className":13043},[36],[38,13045,13046],{"xmlns":40,"display":1043},[42,13047,13048,13076],{},[45,13049,13050,13052,13054,13060,13062,13064,13066,13068,13070,13072,13074],{},[462,13051,7724],{},[87,13053,370],{"stretchy":89},[1065,13055,13056,13058],{},[458,13057,469],{},[458,13059,649],{"mathvariant":5307},[87,13061,370],{"stretchy":89},[458,13063,747],{},[87,13065,98],{"separator":59},[458,13067,757],{},[87,13069,318],{"stretchy":89},[87,13071,318],{"stretchy":89},[87,13073,3459],{},[48,13075,1061],{},[52,13077,13078],{"encoding":54},"\\text{Var}(d_{\\mathbb{S}}(x, y)) \\to 0",[28,13080,13082,13167],{"className":13081,"ariaHidden":59},[58],[28,13083,13085,13088,13094,13097,13140,13143,13146,13149,13152,13155,13158,13161,13164],{"className":13084},[63],[28,13086],{"className":13087,"style":116},[67],[28,13089,13091],{"className":13090},[72,541],[28,13092,7724],{"className":13093},[72],[28,13095,370],{"className":13096},[120],[28,13098,13100,13103],{"className":13099},[72],[28,13101,469],{"className":13102},[72,503],[28,13104,13106],{"className":13105},[508],[28,13107,13109,13132],{"className":13108},[512,513],[28,13110,13112,13129],{"className":13111},[517],[28,13113,13115],{"className":13114,"style":12771},[521],[28,13116,13117,13120],{"style":7358},[28,13118],{"className":13119,"style":530},[529],[28,13121,13123],{"className":13122},[534,535,536,537],[28,13124,13126],{"className":13125},[72,537],[28,13127,649],{"className":13128},[72,5329,537],[28,13130,571],{"className":13131},[570],[28,13133,13135],{"className":13134},[517],[28,13136,13138],{"className":13137,"style":1306},[521],[28,13139],{},[28,13141,370],{"className":13142},[120],[28,13144,747],{"className":13145},[72,503],[28,13147,98],{"className":13148},[130],[28,13150],{"className":13151,"style":135},[134],[28,13153,757],{"className":13154,"style":699},[72,503],[28,13156,10806],{"className":13157},[142],[28,13159],{"className":13160,"style":584},[134],[28,13162,3459],{"className":13163},[563],[28,13165],{"className":13166,"style":584},[134],[28,13168,13170,13173],{"className":13169},[63],[28,13171],{"className":13172,"style":68},[67],[28,13174,1061],{"className":13175},[72],[10,13177,13178,13179,409],{},"Thus, Geodesic distances also concentrate → almost all pairs of points are at distance ",[28,13180,13182,13200],{"className":13181,"translate":32},[31],[28,13183,13185],{"className":13184},[36],[38,13186,13187],{"xmlns":40},[42,13188,13189,13197],{},[45,13190,13191],{},[1187,13192,13193,13195],{},[458,13194,654],{},[48,13196,619],{},[52,13198,13199],{"encoding":54},"\\frac{\\pi}{2}",[28,13201,13203],{"className":13202,"ariaHidden":59},[58],[28,13204,13206,13210],{"className":13205},[63],[28,13207],{"className":13208,"style":13209},[67],"height:1.0404em;vertical-align:-0.345em;",[28,13211,13213,13216,13276],{"className":13212},[72],[28,13214],{"className":13215},[120,1753],[28,13217,13219],{"className":13218},[1187],[28,13220,13222,13268],{"className":13221},[512,513],[28,13223,13225,13265],{"className":13224},[517],[28,13226,13229,13243,13251],{"className":13227,"style":13228},[521],"height:0.6954em;",[28,13230,13231,13234],{"style":2146},[28,13232],{"className":13233,"style":1773},[529],[28,13235,13237],{"className":13236},[534,535,536,537],[28,13238,13240],{"className":13239},[72,537],[28,13241,619],{"className":13242},[72,537],[28,13244,13245,13248],{"style":1822},[28,13246],{"className":13247,"style":1773},[529],[28,13249],{"className":13250,"style":1830},[1829],[28,13252,13253,13256],{"style":2169},[28,13254],{"className":13255,"style":1773},[529],[28,13257,13259],{"className":13258},[534,535,536,537],[28,13260,13262],{"className":13261},[72,537],[28,13263,654],{"className":13264,"style":699},[72,503,537],[28,13266,571],{"className":13267},[570],[28,13269,13271],{"className":13270},[517],[28,13272,13274],{"className":13273,"style":2191},[521],[28,13275],{},[28,13277],{"className":13278},[142,1753],[5284,13280,13282],{"id":13281},"curvature-amplifies-concentration","Curvature Amplifies Concentration",[10,13284,13285],{},"An important insight is: Positive curvature does not eliminate the curse of dimensionality — it reshapes and often strengthens it.",[10,13287,13288],{},"Why?",[3994,13290,13291,13294,13297],{},[3997,13292,13293],{},"The sphere is compact → no radial dispersion",[3997,13295,13296],{},"Curvature forces geodesics to “bend back”",[3997,13298,13299],{},"Volume grows slower than Euclidean space",[10,13301,13302],{},"As a result: Samples are even more tightly concentrated. Angular discrimination becomes harder. nd most directions become indistinguishable",[10,13304,13305,13306,13403],{},"In positively curved spaces: Measure concentrates on equatorial regions, and angles concentrate around ",[28,13307,13309,13326],{"className":13308,"translate":32},[31],[28,13310,13312],{"className":13311},[36],[38,13313,13314],{"xmlns":40},[42,13315,13316,13324],{},[45,13317,13318],{},[1187,13319,13320,13322],{},[458,13321,654],{},[48,13323,619],{},[52,13325,13199],{"encoding":54},[28,13327,13329],{"className":13328,"ariaHidden":59},[58],[28,13330,13332,13335],{"className":13331},[63],[28,13333],{"className":13334,"style":13209},[67],[28,13336,13338,13341,13400],{"className":13337},[72],[28,13339],{"className":13340},[120,1753],[28,13342,13344],{"className":13343},[1187],[28,13345,13347,13392],{"className":13346},[512,513],[28,13348,13350,13389],{"className":13349},[517],[28,13351,13353,13367,13375],{"className":13352,"style":13228},[521],[28,13354,13355,13358],{"style":2146},[28,13356],{"className":13357,"style":1773},[529],[28,13359,13361],{"className":13360},[534,535,536,537],[28,13362,13364],{"className":13363},[72,537],[28,13365,619],{"className":13366},[72,537],[28,13368,13369,13372],{"style":1822},[28,13370],{"className":13371,"style":1773},[529],[28,13373],{"className":13374,"style":1830},[1829],[28,13376,13377,13380],{"style":2169},[28,13378],{"className":13379,"style":1773},[529],[28,13381,13383],{"className":13382},[534,535,536,537],[28,13384,13386],{"className":13385},[72,537],[28,13387,654],{"className":13388,"style":699},[72,503,537],[28,13390,571],{"className":13391},[570],[28,13393,13395],{"className":13394},[517],[28,13396,13398],{"className":13397,"style":2191},[521],[28,13399],{},[28,13401],{"className":13402},[142,1753],", geodesic distances become nearly constant. The curse of dimensionality persists, but manifests as angular collapse instead of radial shell concentration.",{"title":1051,"searchDepth":13405,"depth":13405,"links":13406},2,[13407,13412],{"id":19,"depth":13405,"text":20,"children":13408},[13409,13411],{"id":5286,"depth":13410,"text":5287},3,{"id":9264,"depth":13410,"text":9265},{"id":9967,"depth":13405,"text":9968,"children":13413},[13414,13415,13416,13417],{"id":10207,"depth":13410,"text":10208},{"id":11790,"depth":13410,"text":11791},{"id":12610,"depth":13410,"text":12611},{"id":13281,"depth":13410,"text":13282},false,true,"md",{},"2026-04-10","\u002Fblog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space",{"title":5,"description":12},"blog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space","The curse of dimensionality is not a specific geometric structure, but a collection of phenomena induced by the intrinsic properties of high-dimensional spaces.",[13428,13429],"mathematics","machine-learning","TarTiBbAe4MZykxGesYYZqce6rZbkaVipgaQIAoKyxg",1777090294564]