Calculus of Variations and Geometric Measure Theory
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Q. Xia

Ramified optimal transportation in geodesic metric spaces

created by xia1 on 01 Sep 2021


Published Paper

Inserted: 1 sep 2021
Last Updated: 1 sep 2021

Journal: Adv. Calc. Var.
Volume: 4
Number: 3
Pages: 277-307
Year: 2011

ArXiv: 0907.5596 PDF


An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and 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. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

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