Intuition for spheres in high dimension

Question

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.

Thanks.

Pick a random point $(x_1,\dots,x_D)$ on the surface of the sphere. The expected value of $x_1^2$ is $1/D$, so the point is "probably" around $1/\sqrt D$ away from the equator. — , Mar 21 '18 at 18:59
Related (or not) but perhaps interesting: https://math.stackexchange.com/questions/2644700/whats-new-in-higher-dimensions/2644740#2644740 — Ethan Bolker, Mar 21 '18 at 20:46

Micah · Accepted Answer · 2018-03-21T20:39:22.820

If $(x_1,\dots,x_D)$ is far from the equator, then $|x_D|$ is large. Let $P(C)$ denote the fraction of the sphere where $|x_D|>C$. Then by symmetry, $P(C)$ also gives the fraction of the sphere where $|x_k|>C$ for any $k<D$. But any given point can have at most $\left\lfloor \frac{1}{C^2}\right \rfloor$ coordinates with absolute value larger than $C$, so the regions for all the coordinates can cover the sphere at most $\left\lfloor \frac{1}{C^2}\right \rfloor$ times. As there are $D$ such regions, we have $$ P(C) \leq \frac{1}{D} \left\lfloor \frac{1}{C^2}\right \rfloor $$ which goes to $0$ as $D \to \infty$ for any fixed $C$.

That's pretty neat. Thanks! – SarthakC Mar 21 '18 at 19:48 — SarthakC, Mar 21 '18 at 19:48

score 0 · Answer 2 · answered Mar 21 '18 at 17:00

I dont know much about higher dimensional spheres so I'll leave some links here for reference which are pretty good.

A Breakthrough in Higher Dimensional Spheres | Infinite Series | PBS Digital Studios

Thinking visually about higher dimensions- 3Blue1Brown

Strange Spheres in Higher Dimensions - Numberphile

Intuition for spheres in high dimension

2 Answers2