This is not true! Nearly all of the hypervolume of a high-dimensional n-ball is located near its surface. A popular way of stating this is "If you peel a high-d

This is not true! Nearly all of the hypervolume of a high-dimensional n-ball is located near its surface. A popular way of stating this is "If you peel a high-dimensional orange, there will be almost nothing left." The ratio of "rind" to "pulp" increases very, very quickly. A good intuition for why this happens is that the distance from the center of an n-dimensional hypercube to any of its corners is `r * sqrt(n)`, but the distance from the center of a ball to its surface is always just `r`. So if `r` is fixed, and `n` keeps increasing, the corners get "spikier" and farther from the center. Very quickly, almost all of the hypervolume of a hypercube is located in these remote corners, and almost none of it is in the ball at the center. See https://en.wikipedia.org/wiki/Curse_of_dimensionality https://en.wikipedia.org/wiki/Curse_of_dimensionality