Ramified Optimal Transportation in Geodesic Metric Spaces

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. The optimal transportation problem aims at finding an optimal way to transport a given measure into another with the same mass. In contrast to the well-known Monge-Kantorovich problem (e.g. [1], [6], [7], [14], [15], [18], [20], [22]), the ramified optimal transportation problem aims at modeling a branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor transportation in groups via a nonlinear (typically concave) cost function on mass. Transport networks with branching structures are observable not only in nature as in trees, blood vessels, river channel networks, lightning, etc. but also in efficiently designed transport systems such as used in railway configurations and postage delivery networks. Several different approaches have been done on the ramified optimal transportation problem in Euclidean spaces, see for instance [16], [24], [19], [25], [26], [4], [2], [27], [13], [5], [28], and [29]. Related works on flat chains may be found in [23], [12], [25] and [21]. This article aims at extending the study of ramified optimal transportation from Euclidean spaces to metric spaces. Such generalization is not only mathematically nature but also may be useful for considering specific examples of metric spaces later. By exploring various properties of the metric, we show that many results about ramified optimal transportation is not limited to Euclidean spaces, but can be extended to metric spaces with suitable properties on the metric. Some results that we prove in this article are summarized here: 2000 Mathematics Subject Classification. Primary 49Q20, 51Kxx; Secondary 28E05, 90B06.