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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?

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More can be said: if $A$ is any subset of high-dimensional sphere $S^n$ such that $\operatorname{vol}_nA=\frac12\operatorname{vol}_n S^n$, then the volume is concentrated near the boundary of $A$; in the sense that an $\epsilon$-neighborhood of the boundary contains most of the measure. When $A$ is a hemisphere, you get the statement about equator.

Even more generally: if $f:S^n\to \mathbb R$ is a Lipschitz function, then the pushforward of the Lebesgue measure on $S^n$ is a sharply localized measure on $\mathbb R$ (necessarily, concentrated near the median). Taking $f$ to be the signed distance function to $\partial A$, one recovers the result of previous paragraph.

Suggested reading: Michel Ledoux, The Concentration of Measure Phenomenon. American Mathematical Society, 2001.

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