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arXiv · 0905.3837

On the transport dimension of measures

Qinglan Xia·Anna Vershynina
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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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Qinglan Xia, Anna Vershynina. 2009-05-23. On the transport dimension of measures. https://doi.org/10.1137/090770205

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