yesterday i tried to understand the thing people mean when they say that high-dimensional balls and cubes are "spiky." it seems to me that we can be much more precise than the usual calculations people do here, with very little additional effort. might write a blog post
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in the meantime this blog post is cute and the comment about a multivariate gaussian in high dimensions looking like a donut was super helpful
in high dimensions a multivariate gaussian looks surprisingly similar to a uniform distribution on a sphere
observablehq.com/@tophtucker/th
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the surface of the n-cube [-1, 1]^n is the set of points where at least one coordinate is 1 or -1. the % of points all of whose coordinates are at least x/2n away from 1 or -1 is (1 - x/n)^n ~ e^{-x}. so almost all points have at least one coordinate within O(1/n) of 1 or -1
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this just reflects the idea that a random point in the n-cube is a sequence of n iid samples from the uniform distribution on [-1, 1], so you expect the coordinates to be uniformly distributed and in particular for O(1) of them to be within O(1/n) of the ends