A glimpse into the differential topology and geometry of optimal transport

TL;DR

This paper explores the differential topology and geometry underlying optimal transport phenomena, analyzing Monge maps, Kantorovich measures, and their connections to cost function properties, revealing new geometric insights.

Contribution

It provides a survey of fundamental questions in optimal transport related to geometry and topology, and introduces new connections based on cost function properties.

Findings

Existence and regularity conditions for Monge maps

Uniqueness criteria for Kantorovich measures

Dimension estimates for support of optimal plans

Abstract

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.