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

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, $\{0,\frac{1}{2}-\frac{\sqrt{\frac{3}{7}}}{2},\frac{1}{2},\frac{1}{2}+\frac{\sqrt{\frac{3}{7}}}{2},1\}$

• Looks like the title should read "maximize the product of pairwise distances", rather then "sum of pairwise products"? – Alon Amit Dec 23 '10 at 19:40
• Fixed the title. Not sure of the the tags, though. – Aryabhata Dec 23 '10 at 20:02
• You are right. Thanks. – Ross Millikan Dec 23 '10 at 20:10
• What does 'push out to the ends' mean? 1/2 + 1/(2 sqrt(7)) is less than 0.69, so in this case the points are pushing in to the centre. But +1 for the question – TonyK Dec 23 '10 at 20:13
• Physically, this corresponds to the equilibrium distribution of $n$ points under a pairwise repulsion potential $U(r) = -\log \lvert r \rvert$. People tend to study electrostatic potential instead, $U(r) = 1/r$, but in 1D it feels like there should be a closed-form solution for either case. I don't know what it is, though. – Rahul Dec 23 '10 at 20:19

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 $n-1$.