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For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the optimization problem $$\sup_A \phi(A):=\sup_A \sum_{U\in cc(A)}\int_{U^2} f(x-y)\,dx\,dy,$$ where the sup is taken over all open subsets $A\subset[0,1]^d$. Note that the problem is trivial if $f\ge0$ in which case $\sup_A\phi(A)=\int_{([0,1]^d)^2}f(x-y)\,dx\,dy$.

Has this (or a similar problem) been studied in the literature? It seems to be a rather natural question but I feel that I searched for the wrong keywords. I am mainly interested in the existence of optimal $A$ and their properties (I think I convinced myself that optimal $A$ (if they exist) satisfy $\overline A=[0,1]^d $ due to $f(0)>0$).

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  • $\begingroup$ May I ask for the reason of the downvote? Is the problem too trivial for this site? Would math.stackechange.com be a better resource? $\endgroup$ – Julian Apr 20 '18 at 12:34
  • $\begingroup$ Don't worry about the downvote - you have far more upvotes, and at least I think the problem is really nice. Also the fact that no one has posted a comment answering your question (or posted a proper answer) shows that your problem is challenging at least $\endgroup$ – Dominic van der Zypen Apr 21 '18 at 6:27
  • $\begingroup$ Do you know the answer for $d=1$? $\endgroup$ – Dirk Apr 21 '18 at 6:58
  • $\begingroup$ I guess for $d\ge3$, the $U$ being connected components is no more that relevant: one can always make any of them connected, and also connect it to another one, by means of thin tubes of small measure, without breaking other connections, and with arbitrarily small variation in the value of $\phi$ $\endgroup$ – Pietro Majer Apr 22 '18 at 9:53
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    $\begingroup$ Why do you guess that connecting two connected components with a thin tube only has a small effect on the value of $\phi$? In the case when $f$ is very negative in the long range, connecting distant components should always decrease the value by a lot. $\endgroup$ – Julian Apr 23 '18 at 5:28

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