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Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\in S$, $$\sum_{i=1}^n f(x_i,y_i)\geq \sum_{i=1}^n f(x_i,y_{i+1})$$ where $y_{n+1}\equiv y_1$.

Is there any property (let's call it $\phi$-property), such that the following holds: if a function $f$ is more $\phi$-property than $g$, then for every $S$ that is $g$-cyclically monotone, there exists $S'\supset S$ such that $S'$ is $f$-cyclically monotone.

That is, I am wondering whether there is a property on the function space that generates "larger" cyclically monotone sets.

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  • $\begingroup$ Two brief comments: First, it seems if $S \subset S'$ and $S'$ is $f$-cyclically monotone, then $S$ is also $f$-cyclically monotone, so why include the $S'$ in your statement? And second, I think there may be some hope if you restrict to twice continuously differentiable $f$ and $g$ with some non-degeneracy assumptions using similar ideas to the proof of Theorem 1.2 of McCann, Pass and Warren "Rectifiability of Optimal Transportation Plans". $\endgroup$
    – Steve
    Nov 7, 2022 at 9:50
  • $\begingroup$ Thank you for the comments! $\endgroup$
    – user_XL
    Nov 7, 2022 at 21:20

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