  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-\'Swi\c{e}ch solutions
of their multi-marginal counterparts, both of which involving quadratic cost
functions.
