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 dimensional sphere is proportional to $r^n$. For example the area of a circle (2-sphere) is $\pi r^2$ and the volume of a 3-sphere is $\frac43 \pi r^3$.
This means that the volume grows with a high power of radius, and there is more of that volume near the boundary than near the centre. For a circle, $\frac34$ of the area is closer to the circumference than the centre; for a sphere this becomes $\frac78$ and this increases closer to 1 as the number of dimensions increases. 
With enough dimensions, the outer 10% (by radius), or any small proportion you like, will have more volume than the remainder. Go high enough and the outer 1% of the n-sphere will be responsible for over 99% of the volume.
