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In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.

Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal of the Monge-Kantorovich problem $$ \sup_{\pi \in M(P, Q)} \mathbb{E}_\pi [\Phi(X, Y)] $$ has a comonotone solution.

In the chapter, Galichon assumes that the surplus $\Phi$ is $\mathcal{C}^2$. Hence, I understand that, even if not stated, one should maintan this hypothesis also for the result above.

Now, my question is: do you know another reference for this result, possibly relaxing the $C^2$ assumption?

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  • $\begingroup$ For the sake of clarity, please add the definitions of "supermodular" and "comonotone". $\endgroup$
    – Akira
    Commented Jun 27 at 7:59
  • $\begingroup$ There is a proof right after the statement of the theorem. No differentiability assumption is needed. $\endgroup$
    – Michael Greinecker
    Commented Jun 27 at 8:07
  • $\begingroup$ @MichaelGreinecker thanks! I have seen the proof. But at some point he writes that the results hold by approximation in the case where the domain is the whole $\mathbb{R}^2$. I was afraid the assumption may come into play therein. $\endgroup$
    – Francesco Bilotta
    Commented Jun 27 at 8:38
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    $\begingroup$ The approximation goes from compact rectangles to the plane. Differentiability makes no difference there. $\endgroup$
    – Michael Greinecker
    Commented Jun 27 at 9:29

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