TITLE: Spreading points in the unit interval to maximize the product of pairwise distances
QUESTION [20 upvotes]: This is prompted by question 15312, but moved to the reals.  It must be solved already.  Pick n points $x_i \in [0,1]$ to maximize $\prod_{i < j} (x_i - x_j)$.  A little playing shows you don't want them evenly distributed-they need to push out to the ends.  With four points, Alpha says to use $\{0,\frac{1}{2}\pm\frac{1}{2\sqrt{5}},1\}$ and with five, $\{0,\frac{1}{2}-\frac{\sqrt{\frac{3}{7}}}{2},\frac{1}{2},\frac{1}{2}+\frac{\sqrt{\frac{3}{7}}}{2},1\}$

REPLY [17 votes]: These points are known as Fekete points. A general Fekete problem is to maximize the product 
$$\max_{z_1,...,z_n\in E}\prod\limits_{\quad 1\leq i < j \leq n}|z_i-z_j|$$
where $E\subset \mathbb C$.
In case $E=[-1,1]$, there is a unique solution and the corresponding points  coincide with the zeros of $(1-x^2)P'_{n-1}(x)$, where $P_{n-1}$ is the Legendre polynomial of degree $n-1$.
I cannot give a precise reference at the moment, but this can be probably found in Szegő's book on orthogonal polynomials.