For integers $n\geq k>0$, let $f$ be the following quadratic form: $$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$ Is it true that the minimum of $f$ over the unit simplex is attained at $(1/n,\ldots,1/n)$? Where the unit simplex is the set $\{x\in\mathbb R^n:x_i\geq 0\forall i,\ \sum x_i=1\}$.
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$\begingroup$ This question is related to a former question of mine: mathoverflow.net/q/321720/85550 $\endgroup$– Ron PCommented Jan 30, 2019 at 10:43
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$\begingroup$ Well, it is not generally true. I have a counterexample for k even and n a multiple of k+1. Eg, (1/2,1/2,0,0,0). $\endgroup$– Ron PCommented Jan 30, 2019 at 15:58
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I wrote what I thought to be a proof without seing your counter example. Yes it seems to be false in general. However, it is true for $k=1$ or $k=n$ and I think that if the minimum is achieved on the interior of the simplex (so that all the $x_i$'s are positive) then by perturbing then one can show that necessarily it is the point $(1/n, \dots, 1/n)$.
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1$\begingroup$ Thanks! k=1, the function is $\sum x_i^2$ which has a unique minimum at (1/n,...,1/n). k=n, the function is constant 1. It might be true for k=2,3 as well, but I'm not sure. $\endgroup$– Ron PCommented Jan 31, 2019 at 11:16