# How can it be that the volume of a d dimensional ball is concentrated on two different equators?

In the answer to Explanation for this event on a high-dimensional dataset it is stated that: "almost all the surface area of a sphere in d -dimensional Euclidean space Ed is concentrated around its equator." and it is supposed that the equator is relative to an arbitrary point on the sphere.

What if a different random point on the sphere is now selected as the "north pole"? My understanding is that a new equator is now defined where the volume of the ball is concentrated.

This seems to create a paradox: how can it be that the ball's volume is concentrated on two different equators simultaneously? Or is it alternatively that the selection of an arbitrary "north pole" subjectively creates the corresponding concentration at the equator?

• Welcome to CrossValidated, and thank you for a very nice first question! I took the liberty of adding a few paragraph breaks. I also linked here from the original question, let's see whether we get any answers. (I'm not completely sure this would not be better asked at Math.SE...) Commented Feb 13, 2023 at 9:50
• An equator of a hypersphere here seems to be defined relative to a point (in effect the North or South pole) or an axis through that point and the centre, and most of the hyper-surface-area is almost as far away from that axis as possible. In the linked question there is one specific point. In your question there seem to be two equators so two points; in high dimensions, this is saying that most other points are concentrated near both the resulting equators. The 3D analogy does not work well here in thinking about this. Commented Feb 13, 2023 at 10:44
• There's no paradox: the tubular neighborhoods of any two equators have substantial regions of intersection. The 3D analogy is perfectly good--the intersection is a pair of disconnected neighborhoods forming a buffer around the one dimensional spherical intersection of the equators--but in $d\gg 1$ dimensions, the intersection is a buffer around a $d-2$ dimensional subset of the $d-1$-sphere and that intersection becomes substantial as $d$ grows.
– whuber
Commented Feb 13, 2023 at 14:46
• Related: mathoverflow.net/q/210291/136862 Commented Jun 13, 2023 at 0:05