Is there an analog of Maxwell's theorem on the unit ball? Maxwell's theorem says that any product measure on $\mathbb{R}^n$ which is invariant to orthogonal matrices (meaning that the measure of $A$ and the image of $A$ under any orthogonal map $Q$ is the same) is a Normal distribution of the form $N(0, \kappa I_n)$ for $\kappa > 0$.
Is there an analogous statement when instead of $\mathbb{R}^n$ we make the space the unit ball in $\mathbb{R}^n$?
 A: The only product measure on the unit $n$-ball $B^n$ which is orthogonally invariant is the Dirac measure at the origin. (Edit: as with the usual Maxwell's theorem we need $n \ge 2$ here because the $n \le 1$ cases are degenerate.)
You can see this by asking what the support of such a measure can be. As a product measure with support contained in $B^n$, its support must be contained in a product of intervals $[a, b]$ (given by intervals containing the supports of its factors) contained in $B^n$. So its support must be contained in the largest $n$-cube inscribable in $B^n$. This cube takes the form $[-s, s]^n$ where $ns^2 = 1$, so $s = \frac{1}{\sqrt{n}}$.
But by orthogonal invariance the support of the measure must be a union of spheres (edit: centered at the origin). So the support must be contained in the largest ball inscribable in this $n$-cube. This is a ball of radius $s = \frac{1}{\sqrt{n}}$!
Iterating this argument shrinks the radius by a factor of $\frac{1}{\sqrt{n}}$ each time; we conclude that the measure can only be supported at the origin.
