On the transport dimension of measures

TL;DR

This paper introduces the concept of transport dimension for probability measures using ramified optimal transportation, establishing bounds with classical dimensions and defining a metric that relates transport dimension to geometric distance.

Contribution

It defines the transport dimension via ramified optimal transportation, relates it to Minkowski and Hausdorff dimensions, and introduces a new metric called the dimensional distance.

Findings

Transport dimension is bounded above by Minkowski dimension.

Transport dimension is bounded below by Hausdorff dimension.

The dimensional distance measures the transport dimension as the distance to finite atomic measures.

Abstract

In this article, we define the transport dimension of probability measures on $\mathbb{R}^m$ using ramified optimal transportation theory. We 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 on $\mathbb{R}^m$. This metric gives a geometric meaning to the transport dimension: with respect to this metric, we show that the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.