In particular, I'm interested in the property that the surface area of a sphere in D dimensions gets concentrated near the equator. I know it can be shown with some integrals, for example done in Section 1.2.5 here.
What I want to know, if anyone can give me a short relatively verbal argument for this. It's a point that I need to make in a talk that I'm giving, and I'd like to give some intuition for why this result is true, but going into an integral or something would detract away too much from my main content. A little math is fine (It's going to be an applied math/physics audience), but something which wont take up too much time to explain if possible.