Monge solutions and uniqueness in multi-marginal optimal transport: weaker conditions on the cost, stronger conditions on the marginals

TL;DR

This paper introduces a new general condition on cost functions in multi-marginal optimal transport that guarantees uniqueness and Monge solutions, even with weaker assumptions on the marginals than previously required.

Contribution

It presents a broader condition on cost functions ensuring Monge solutions and uniqueness, extending beyond the twist on splitting sets condition.

Findings

New condition guarantees Monge solutions with weaker marginal assumptions

Several cost functions satisfy the new condition but violate previous conditions

Results include examples of cost functions with guaranteed uniqueness and Monge solutions

Abstract

We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to local coordinates. When only the first marginal is assumed to be absolutely continuous, our condition is equivalent to the twist on splitting sets condition found in [23]. In addition, it is satisfied by the special cost functions in our earlier work [32, 33], when absolute continuity is imposed on certain other collections of marginals. We also present several new examples of cost functions which violate the twist on splitting sets condition but satisfy the new condition introduced here; we therefore obtain Monge solution and uniqueness results for these cost functions, under regularity conditions on an appropriate subset of the marginals.