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High dimension
In high dimension, it's really difficult to stay local.
In high dimensions, all cases are edge cases
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Example
See this interactive app in R Shiny on the wiki/Curse of Dimensionality.
Circle example: The circle fills up most of the area in the square, in fact it takes up exactly <math>\pi</math> out of 4 which is about 78%. In three dimensions we have a sphere and a cube, and the ratio of sphere volume to cube volume is a bit smaller, <math>\frac{4\pi}{3}</math> out of a total of 8, which is just over 52%