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DECOMPOSITION OF GEODESICS IN THE WASSERSTEIN SPACE AND THE GLOBALIZATION PROBLEM
"... Abstract. We will prove a decomposition for Wasserstein geodesics in the following sense: let (X, d, m) be a nonbranching metric measure space verifying CDloc(K, N) or equivalently CD ∗ (K, N). Then every geodesic µt in the L 2Wasserstein space, with µt ≪ m, is decomposable as the product of two d ..."
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Abstract. We will prove a decomposition for Wasserstein geodesics in the following sense: let (X, d, m) be a nonbranching metric measure space verifying CDloc(K, N) or equivalently CD ∗ (K, N). Then every geodesic µt in the L 2Wasserstein space, with µt ≪ m, is decomposable as the product of two densities, one corresponding to a geodesic with support of codimension one verifying CD ∗ (K, N − 1), and the other associated with a precise one dimensional measure, provided the length map enjoys local Lipschitz regularity. The motivation for our decomposition is in the use of the component evolving like CD ∗ in the globalization problem. For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining therefore the global CD(K, N) for µt. The result can be therefore interpret as a globalization theorem for CD(K, N) for this class of optimal transportation, or as a “selfimproving property ” for CD ∗ (K, N). Assuming more regularity, namely in the setting of infinitesimally strictly convex metric measure space, the one dimensional density is the product of two differentials giving more insight on the density
FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS
, 2010
"... In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing i ..."
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In this series of lectures we introduce the MongeKantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for MongeAmpère type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.
MONGE PROBLEM IN METRIC MEASURE SPACES WITH RIEMANNIAN CURVATUREDIMENSION CONDITION
"... Abstract. We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X, d, m) enjoying the Riemannian curvaturedimension condition RCD ∗ (K, N), with N < ∞. For the first marginal measure, we assume that µ0 ≪ m. As a corollary, we obtain that the ..."
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Abstract. We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X, d, m) enjoying the Riemannian curvaturedimension condition RCD ∗ (K, N), with N < ∞. For the first marginal measure, we assume that µ0 ≪ m. As a corollary, we obtain that the Monge problem and its relaxed version, the MongeKantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for dcyclically monotone sets: neglecting a set of zero mmeasure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics. Contents
DOI 10.1007/s1337301100027 Optimal transportation, topology and uniqueness
"... © The Author(s) 2011. This article is published with open access at SpringerLink.com Abstract The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smoo ..."
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© The Author(s) 2011. This article is published with open access at SpringerLink.com Abstract The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation. It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and PierreAndre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International
Bull. Math. Sci. DOI 10.1007/s1337301100027 Optimal transportation, topology and uniqueness
, 2010
"... © The Author(s) 2011. This article is published with open access at SpringerLink.com Abstract The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smoo ..."
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© The Author(s) 2011. This article is published with open access at SpringerLink.com Abstract The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation. It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and PierreAndre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International
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MEASURE RIGIDITY OF RICCI CURVATURE LOWER BOUNDS
"... Abstract. The measure contraction property, MCP for short, is the weakest among the Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its sup ..."
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Abstract. The measure contraction property, MCP for short, is the weakest among the Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Radon measure m over Rd is such that (Rd,  · ,m) verifies a weaker variant of MCP, then its support spt(m) must be convex and m has to be absolutely continuous with respect to the relevant Hausdorff measure of spt(m). This result is then used as a starting point to investigate the rigidity of MCP in the metric framework. We introduce the new notion of reference measure for a metric space and prove that if (X, d,m) is essentially nonbranching and verifies MCP, and µ is an essentially nonbranching MCP reference measure for (spt(m), d), then m is absolutely continuous with respect to µ, on the set of points where an inversion plan exists. As a consequence, an essentially nonbranching MCP reference measure enjoys a weak type of uniqueness, up to densities. We also prove a stability property for reference measures under measured GromovHausdorff convergence, provided an additional uniform bound holds. In the final part we present concrete examples of metric spaces with reference measures, both in smooth and nonsmooth setting. The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group. 1.