A note on optimal transport and Monge-Amp\`ere geometry

TL;DR

This paper explores a new link between a pseudo-Riemannian approach to optimal transport and Monge-Ampère geometry, demonstrated through an example in geophysical fluid dynamics.

Contribution

It introduces a novel connection between optimal transport geometry and Monge-Ampère equations, expanding the theoretical understanding of both fields.

Findings

Established a correspondence between pseudo-Riemannian optimal transport and Monge-Ampère geometry

Applied the theoretical framework to a geophysical fluid dynamics example

Provided insights into the geometric structure underlying optimal transport problems

Abstract

We identify a novel connection between a recently introduced pseudo-Riemannian framework for optimal mass transport and the geometry of Monge-Amp\`ere equations. We show this correspondence by application to an example from geophysical fluid dynamics.