As the dimension of a hypersphere increases, where are a set of uniformly distributed points more likely to be?

Question

Around the core? Equator? Surface?

I am having trouble comprehending how these things relate to each other in higher dimensions.

Answer 1

The question isn't as trivial as it may sound. Much depends on what is mean by "where". As, MvG points out in the comments, when comparing equal volumes anywhere, they should have the same number of points in expectation.

However, there are other ways to frame the "where" question. Near the center or far from it, for example. This latter question is not comparing equal volumes.

Consider the unit ball B . Consider the ball $B_{1-\epsilon}$ within it or radius $1-\epsilon$. The volume of $B_{1-\epsilon}$ is $\propto ({1-\epsilon})^n$. No matter how small $\epsilon$ is, this is exceedingly small as $n \to \infty$. That is, most of the volume is close to the boundary.

What about distance from equator when that is considered for answering the "where". As $n \to \infty$, of the volume will lie within $\epsilon$ of the equatorial hyperplane.

Take a look at this document to find everything you wanted to know about high dimensional spheres.