Remarks on multi-marginal symmetric Monge-Kantorovich problems

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 ($m\ge 2$). 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-'Swi\c{e}ch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.