Ramified optimal transportation in geodesic metric spaces

TL;DR

This paper extends the theory of ramified optimal transportation from Euclidean spaces to general geodesic metric spaces, exploring existence, properties, and introducing new concepts like transport dimension and dimensional distance.

Contribution

It generalizes ramified optimal transport theory to geodesic metric spaces and introduces the transport dimension and dimensional distance, linking geometric measure theory with optimal transport.

Findings

Transport dimension is bounded by Minkowski and Hausdorff dimensions.

Existence and behavior of optimal transport paths depend on metric properties.

Introduces a new metric, the dimensional distance, with geometric interpretation.

Abstract

An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.