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Remarks on multi-marginal symmetric Monge-Kantorovich problems

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  • Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Święch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.
    Mathematics Subject Classification: Primary: 49J30, 49K30, 49J52; Secondary: 58C35, 58E17, 91B68.


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