Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:

$D=\displaystyle \prod_{1\leq i < j \leq n} \overline{P_iP_j}$

I wonder...

1) What is the probability density function for $D$? What is the expected value of $D$?

2) When the $P_i$ are equally spaced, we know $D=n^{n/2}$. Is this an absolute maximum for $D$?