Calculus of Variations and Geometric Measure Theory
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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

ArXiv: 0905.3837 PDF

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.

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