Box distance and observable distance via optimal transport

TL;DR

This paper explores the box and observable metrics on metric measure spaces, representing them through optimal transport plans and proving the existence of such plans for both metrics.

Contribution

It provides a novel representation of Gromov's box and observable metrics using transport plans and establishes the existence of optimal transport plans for these metrics.

Findings

Representation of metrics via transport plans

Existence of optimal transport plans for both metrics

Enhanced understanding of metric measure space geometry

Abstract

On the set of all metric measure spaces, we have two important metrics, the box metric and the observable metric, both introduced by M. Gromov. We obtain the representation of these metrics by using transport plan. In addition, we prove the existence of optimal transport plans of these metrics.