# Intuition for spheres in high dimension

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.

• surface is concentrated near the equator? – Peter Franek Mar 21 '18 at 16:30
• @PeterFranek: My bad. I meant surface area – SarthakC Mar 21 '18 at 16:53
• 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. – Rahul Mar 21 '18 at 18:59
• Related (or not) but perhaps interesting: math.stackexchange.com/questions/2644700/… – Ethan Bolker Mar 21 '18 at 20:46

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 $C$.