## Q. Xia - A. Vershynina

# On the transport dimension of measures

created by xia1 on 01 Sep 2021

[

BibTeX]

*Published Paper*

**Inserted:** 1 sep 2021

**Last Updated:** 1 sep 2021

**Journal:** SIAM J. MATH. ANAL

**Volume:** 41

**Number:** 6

**Pages:** 2407-2430

**Year:** 2010

**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.