*Published Paper*

**Inserted:** 14 sep 2009

**Last Updated:** 6 aug 2010

**Journal:** Manuscripta Math.

**Volume:** 133

**Pages:** 197-224

**Year:** 2010

**Abstract:**

We study a new class of distances between Radon measures similar to those studied in a recent paper of Dolbeault-Nazaret-SavarÃ© DNS. These distances (more correctly pseudo-distances because can assume the value $+\infty$) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in DNS) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in $R^d$ with finite moments and the set of measures defined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness.

**Keywords:**
generalized Wasserstein distance, mobility function

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