Let's n points $M_k$ on the unit circle. Find the position of these points such that $$P_n = \prod_{1 \le k \le l \le n}M_kM_l$$ (product of all the distances between the points) is maximum.
With the Hadamard inequality, we can proove that $P_n \le n^{\frac{n}{2}}$.
If the points are the $n$-roots of unity (or the image by a rotation of the $n$-roots of unity), we can proove that the inequality is a equality.
My question : how to proove that the $n$-root of unity are the only (modulo a rotation) configuration that realize the maximum ?