Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces

TL;DR

This paper investigates the properties of Kantorovich potentials in metric measure spaces and extends Brenier's theorem to broader non-branching geodesic spaces, establishing conditions for the existence of optimal transport maps.

Contribution

It introduces slope concepts along curves and geodesics and generalizes Brenier's theorem to non-branching geodesic metric spaces.

Findings

Defined slopes along curves and geodesics in metric spaces

Proved existence of optimal transport maps under new conditions

Extended Brenier's theorem to non-branching geodesic spaces

Abstract

We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.