Relative Optimal Transport

TL;DR

This paper introduces a theory of optimal transport relative to a subset, extending classical concepts like Wasserstein distance and duality to compare measures of different total variation.

Contribution

It develops a comprehensive framework for relative optimal transport, including duality theorems and connections to Lipschitz functions, with potential independent interest in Riesz cones.

Findings

Established the existence of optimal solutions in the relative transport problem.

Derived relative versions of key duality theorems in optimal transport.

Connected measures with Lipschitz function spaces via Riesz-Markov-Kakutani theorem.

Abstract

We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest.