The Monge Problem for distance cost in geodesic spaces

TL;DR

This paper investigates the Monge optimal transport problem in geodesic metric spaces, reducing it to 1D problems along geodesics under certain regularity conditions, and explores the properties of optimal transport maps.

Contribution

It introduces new assumptions ensuring regularity of conditional probabilities, enabling the construction of transport maps in geodesic spaces with distance cost.

Findings

Reduction of the transport problem to 1D along geodesics

Regularity conditions imply existence of transport maps

dL-cyclical monotonicity is not sufficient for optimality

Abstract

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X,dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem {\pi} which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1- dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.