Given a region $S$ in the Euclidean plane, let $X$ and $Y$ be random points in $S$, independent and uniformly distributed.
If the area of $S$ is given (let's say it is 1), what is the shape of $S$ that minimizes the variance of distances, $$ \operatorname{Var}(d(X,Y)), $$ where $d$ is the Euclidean distance?
My guts says is a ball (in 2D). But I don't know how to prove it.
A more simple question is what shape will minimize the expectation of $d(X,Y)$. This time again I think its a ball, but I have some more sense.
Not a proof, but a direction, is that in any shape that is not a ball, there is a point that we can get closer to the center of mass of the group - hence decreasing the average distance from the rest of the points.