I know that (as the dimension increases) the volume of a hypersphere concentrates near the equators and near the surface, and that the surface area (SA) concentrates near the equators as well.
What more can be said?
An equator of an n-sphere is an (n-1)-sphere... Does the volume/SA concentrate near the equator(s) of the equator(s)? (and so on)?
In general, are there some other well defined regions around which volume/SA concentrate in high dimensional spheres?