On the transport dimension of 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.

24 pages, 5 figures. Rewrite some paragraphs to section 2; correcting some typing error

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