Optimal transport maps on Alexandrov spaces revisited

TL;DR

This paper provides an alternative proof demonstrating the existence and uniqueness of optimal transport maps in certain Alexandrov spaces with curvature bounds, extending the understanding of optimal transport in geometric analysis.

Contribution

It offers a new proof for the existence and uniqueness of optimal transport maps in Alexandrov spaces with curvature bounds, enhancing theoretical understanding.

Findings

Existence of unique optimal transport plans in Alexandrov spaces

Optimal plans are induced by maps in these spaces

Extension of optimal transport theory to non-smooth geometric settings

Abstract

We give an alternative proof for the fact that in $n$-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely $(n-1)$-unrectifiable starting measure, and that this plan is induced by an optimal map.