A class of multi-marginal c-cyclically monotone sets with explicit c-splitting potentials

TL;DR

This paper introduces a new, easier-to-verify condition for constructing c-splitting potentials in multi-marginal optimal transport, extending classical convex analysis and applicable to certain cost functions and set margins.

Contribution

It provides an explicit construction method for c-splitting potentials under a new sufficient condition for multi-marginal c-cyclically monotone sets.

Findings

The new condition is sufficient for explicit c-splitting potential construction.

The condition is easier to verify than multi-marginal c-cyclic monotonicity.

When margins are one-dimensional, the condition characterizes c-splitting sets.

Abstract

Multi-marginal optimal transport plans are concentrated on c-splitting sets. It is known that, similar to the two-marginal case, c-splitting sets are c-cyclically monotone. Within a suitable framework, the converse implication was very recently established by Griessler. However, for an arbitrary cost c, given a multi-marginal c-cyclically monotone set, the question whether there exists an analogous explicit construction to the one from the two-marginal case of c-splitting potentials is still open. When the margins are one-dimensional and the cost belongs to a certain class, Carlier proved that the two-marginal projections of a c-splitting set are monotone. For arbitrary products of sets equipped with cost functions which are sums of two-marginal costs, we show that the two-marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of c-splitting potentials. Our condition is, in principle, easier to verify than the one of multi-marginal c-cyclic monotonicity. Various examples illustrate our results. We show that, in general, our condition is sufficient; however, it is not necessary. On the other hand, we conclude that when the margins are one-dimensional equipped with classical cost functions, our condition is a characterization of c-splitting sets and extends classical convex analysis.