Where are most points in a uniformly distributed high-dimensional ball? Should they be close to the middle (origin) or close its surface?
 A: As pointed out by @Xi'an, the OP's question is actually about a uniform distribution on the $n$-dimensional ball of radius $r$, the set of points at distance no more than $r$ from the center of the ball, and not about a uniform distribution on the $n$-dimensional hypersphere which is the surface of the ball (the set of points at distance exactly $r$ from the center). Note that it is being assumed that the joint density of the $n$ random variables has constant value $V^{-1}$ where $V$ is the volume of the ball. This is not the same as assuming that the distance of the random point is uniformly distributed on $[0,r]$ (or $[0,r)$ for those who do not want to include the surface of the hypersphere).
Almost the entire volume of a $n$-dimensional ball lies close to the surface. This is because $V$ is proportional to the $n$-th power of the radius of the ball, and $r^n$ is a very rapidly increasing function.  Even in $3$-space, $\frac 78 = 1 - \left(\frac 12\right)^3$th of the volume lies closer to the surface than to the origin, and this fraction gets closer and closer to $1$ as $n$ increases.  Turning the calculation  around, for a fixed proportion $\alpha$, say $\alpha=0.95$, $100\alpha\%$ of the volume lies in a shell of inner radius $\sqrt[n]{\alpha}r$ and outer radius $r$ and so $1-\sqrt[n]{\alpha}$, the relative thickness of the shell, decreases towards $0$ with increasing $n$ for any choice of $\alpha \in (0,1)$.
