On the Transport Dimension of Measures

Abstract. In this article, we define the transport dimension of probability measures on Rm 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 Rm. 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 the distance from it to any finite atomic probability measure.