concentration of volume of hypersphere

Question

I am reading about features of volume of hyperballs, where I see two theorems,

1. Most of the volume of the d-dimensional ball of radius r is contained in an annulus of width $O(r/d)$ near the surface.

2. Most of the volume of the upper hemisphere of the d-dimensional ball is below the plane $x_1 = c/(d-1)^{1/2}$

Namely, most of volume of hyperball is near the surface, and, most of volume of hyperball is near the equator.

How can these two happen at the same time? I find it very hard to have an image that can boil these two together. Can someone explain why this isn't a contradiction?

Currently my guess is you don't know what "north pole" you have for reference when you pick the equator, so the two theorems can be resolved in the way that most of volume is only in the proportion of annulus near the surface which overlaps the equator currently considered (by picking a point as north pole). I am not sure whether this is the reason.

Thank you for your suggestions.

Answer 1

Well, both of these are true for the ordinary sphere in three dimensions as well, just not as much. More of the volume of a sphere is near the surface than at any other depth. To put it pictorally, if you were to think of a sphere in layers like an onion, the layers near the surface contain more volume simply because they're bigger.

On the other hand, if you take an onion and cut it into slices, the slices near the middle—the equator, in other words—are larger because the sphere is wider there.

What happens in higher dimensions is generally analogous, but because the dimensions are higher, the effect is greater. For instance, the layers in a hypersphere of dimension $n$ increase in size as $r^{n-1}$; in an ordinary sphere, that's $r^2$, but in (say) an eight-dimensional hypersphere, that would be $r^7$.