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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    $\begingroup$ Looks like the title should read "maximize the product of pairwise distances", rather then "sum of pairwise products"? $\endgroup$ – Alon Amit Dec 23 '10 at 19:40
  • $\begingroup$ Fixed the title. Not sure of the the tags, though. $\endgroup$ – Aryabhata Dec 23 '10 at 20:02
  • $\begingroup$ You are right. Thanks. $\endgroup$ – Ross Millikan Dec 23 '10 at 20:10
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    $\begingroup$ 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 $\endgroup$ – TonyK Dec 23 '10 at 20:13
  • $\begingroup$ 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. $\endgroup$ – 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 $n-1$.

I cannot give a precise reference at the moment, but this can be probably found in Szegő's book on orthogonal polynomials.

  • $\begingroup$ Apparently, these are also called Gauss-Labatto points. $\endgroup$ – Aryabhata Dec 23 '10 at 21:34
  • $\begingroup$ I think Gauss-Labatto is specific to the case of one real dimension. (Which, obviously, is what the OP asked about.) And I think (I maybe wrong on this count) their constructions/motivations are different. $\endgroup$ – Willie Wong Dec 23 '10 at 22:01
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    $\begingroup$ @Willie: Yeah, I was trying to find a reference for Andrey when I came across this. Just thought it might be interesting and might help find a reference. $\endgroup$ – Aryabhata Dec 23 '10 at 22:51
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    $\begingroup$ Here's a good reference, I think: Saff & Totik books.google.com/… $\endgroup$ – Hans Lundmark Dec 24 '10 at 15:34
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    $\begingroup$ @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). $\endgroup$ – Hans Lundmark Dec 27 '10 at 15:02

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