On the transport dimension of measures
Skip to main content
Open Access Publications from the University of California

## On the transport dimension of measures

• Author(s): Xia, Qinglan
• Vershynina, Anna
• et al.

## Published Web Location

https://arxiv.org/pdf/0905.3837.pdf
No data is associated with this publication.
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.

Item not freely available? Link broken?
Report a problem accessing this item