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

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    @Rahul: Logarithmic potentials are well studied. See the book by Saff & Totik in my comment to Andrey's answer.2010-12-24

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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.

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    @Willie: What I meant is that I'm under the impression that it's one of the standard references on the subject, but I haven't read it myself (unfortunately). I read a tiny bit about this in Percy Deift's *Orthogonal Polynomials and Random Matrices* years ago, and the book by Saff & Totik was recommended there, along with *Foundations of Modern Potential Theory* by N. S. Landkof (which I haven't read either).2010-12-27