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This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal… (More)

This paper continues the investigation of " Wasserstein-like " transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics W N on the d-dimensional discrete torus T d N with mesh size 1 N converge, when N → ∞, to the standard 2-Wasserstein distance W 2 on the continuous torus in the sense of… (More)

In this paper we introduce a new transportation distance between non-negative measures inside a domain Ω. This distance enjoys many nice properties, for instance it makes the space of non-negative measures inside Ω a geodesic space without any convexity assumption on the domain. Moreover we will show that the gradient flow of the entropy functional Ω [ρ… (More)

We prove that any Kantorovich potential for the cost function c = d 2 /2 on a Riemannian manifold (M, g) is locally semiconvex in the " region of interest " , without any compactness assumption on M , nor any assumption on its curvature. Such a region of interest is of full µ-measure as soon as the starting measure µ does not charge n − 1-dimensional… (More)

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