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I am trying to prove that the global maxima of the following function $f_n(x_1,\ldots,x_n):=\exp(-\sum_{i=1}^n x_i ^2)\prod_{1\leq i (note the second product does not have the restriction $i\lt j$) satisfy the following symmetry property: If $x^*=(x_1^*,\ldots,x_n^*)$ is a global maximum of $f_n$, then the set $\{x_1^*,\ldots,x_n^*\}$ is symmetric around the origin in the sense that if $a\in \{x_1^*,\ldots,x_n^*\}$, then $-a\in \{x_1^*,\ldots,x_n^*\}$ too.

I expect this to be true, but I have no proof. I have checked it numerically for $n=2,3,4$, and I am trying to come up with a nice proof of this fact without much success.

Any suggestions would be appreciated!

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    @oenamen: It is related to graph colorings. See http://www.sciencedirect.com/science/article/pii/037026939390075S2012-04-23

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Since $f_n(x_1,\ldots,x_n) = f_n(-x_1,\ldots,-x_n)$, if $x^*$ is a critical point, then so is $-x^*$. The symmetry property follows from this.

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    @Enrique: It doesn't, I misunderstood your question. Sorry.2012-04-10