Remarks on multi-marginal symmetric Monge-Kantorovich problems
Nassif Ghoussoub Bernard Maurey
Discrete & Continuous Dynamical Systems - A 2014, 34(4): 1465-1480 doi: 10.3934/dcds.2014.34.1465
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.
keywords: $m$-cyclically antisymmetric functions. Monge-Kantorovich duality $m$-cyclically monotone vector fields Mass transport

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