Why is volume of a high-D ball concentrated near its surface? I came across the following sentence while reading a book on applied math:

Volume of a high dimensional unit ball is concentrated near its
  surface and is also concentrated at its equator.

This is from book's introduction, and I believe the sentence will be explained at some later point in the book. However, it is hard to me to accept it on intuition level. Can some of you explain this sentence to me, in a sort of laymen-style, if possible?

The volume of an $n$-dim ball with radius $r$ is of the form $A_n r^n=V_{n,r}$ (see the expression by @Jack).
Let's fix the radius of the ball to unity, and plot what fraction of volume (of the ball with radius 1) is within a smaller (concentric) ball with radius $r$ ($0\leq r \leq 1$).
This fraction is $\frac{V_{n,r}}{V_{n,1}} = \frac{A_n r^n}{A_n 1^n} = r^n$. For different $n$ the plots looks like the following ($y$-axis is $\frac{V_{n,r}}{V_{n,1}}$).

The observation is that for large $n$ (see bottom-most red plot) the fraction is almost zero until $r$ gets closer to 1. This means that the most of the volume of the ball is within a thin shell closer to its surface.
