Metric conditions that guarantee existence and uniqueness of Optimal Transport maps

TL;DR

This paper establishes metric conditions, including a local metric twist and non-branching assumptions, that ensure the existence and uniqueness of optimal transport maps in measure spaces.

Contribution

It introduces a nonsmooth local metric twist condition and proves existence and uniqueness of optimal transport maps under this and non-branching assumptions.

Findings

Existence and uniqueness under local metric twist condition

Existence and uniqueness for quadratic cost in non-branching spaces

New metric conditions generalizing previous results

Abstract

We investigate metric conditions that allow to prove existence and uniqueness of a map solving the Monge problem between two marginals in a metric (measure) space, proving two main results. Firstly, we introduce a nonsmooth version of the Riemannian twist condition that we call local metric twist condition, showing, under this assumption on the cost function, existence and uniqueness of optimal transport maps. Secondly, we prove the same result for cost equal to $d^2$ in a metric space $(X, d)$ satisfying a quantitative non-branching assumption, that we call locally-uniformly non-branching.