On the transport dimension of measures

Details On the transport dimension of measures On the transport dimension of measures

2009-05-23

arxiv.org/abs/0905.3837v3

Mathematics

Optimization and Control

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

Scientific paper

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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