Remarks on multi-marginal symmetric Monge-Kantorovich problems

TL;DR

This paper explores the connections between symmetric multi-marginal Monge-Kantorovich problems and classical solutions like Brenier and Gangbo-Święch, focusing on quadratic cost functions and their implications for vector field decompositions.

Contribution

It establishes relationships between symmetric transport problems and classical solutions, extending the understanding of multi-marginal optimal transport with quadratic costs.

Findings

Relates symmetric Monge-Kantorovich problems to Brenier solutions.

Connects these problems to Gangbo-Święch solutions.

Provides insights into vector field polar decompositions.

Abstract

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.