Results 1  10
of
54
Ricci curvature for metricmeasure spaces via optimal transport
 ANN. OF MATH
, 2005
"... We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of proba ..."
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Cited by 234 (10 self)
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We define a notion of a measured length space X having nonnegative NRicci curvature, for N ∈ [1, ∞), or having ∞Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured GromovHausdorff limits. We give geometric and analytic consequences.
Optimal transportation on noncompact manifolds
"... In this work, we show how to obtain for noncompact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian dis ..."
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Cited by 43 (12 self)
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In this work, we show how to obtain for noncompact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction. 1
Optimal transport and Perelman’s reduced volume
, 2008
"... We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume. ..."
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Cited by 29 (1 self)
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We show that a certain entropylike function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
Finsler interpolation inequalities
, 2009
"... We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a ne ..."
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Cited by 19 (5 self)
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We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.
Optimal transport under nonholonomic constraints
 Transactions of the Amer. Math. Soc
"... Abstract. We study Monge’s optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantl ..."
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Cited by 14 (3 self)
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Abstract. We study Monge’s optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2generating distribution and cost given by d 2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane. 1.
Mass Transportation on SubRiemannian Manifolds
"... We study the optimal transport problem in subRiemannian manifolds where the cost function is given by the square of the subRiemannian distance. Under appropriate assumptions, we generalize BrenierMcCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute ..."
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Cited by 14 (2 self)
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We study the optimal transport problem in subRiemannian manifolds where the cost function is given by the square of the subRiemannian distance. Under appropriate assumptions, we generalize BrenierMcCann’s Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the MongeAmpère equation. 1
GENERALIZED RICCI CURVATURE BOUNDS FOR THREE DIMENSIONAL CONTACT SUBRIEMANNIAN MANIFOLDS
, 903
"... Abstract. Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property. 1. ..."
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Cited by 14 (5 self)
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Abstract. Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property. 1.
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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Cited by 13 (4 self)
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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.