Spreading points in the unit interval to maximize the product of pairwise distances
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.