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229
On the geometry of metric measure spaces
 II, ACTA MATH
, 2004
"... We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Amo ..."
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Cited by 259 (8 self)
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We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Among others, we show that Curv(M, d,m) ≥ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, Curv(M, d,m) ≥ K if and only if RicM (ξ, ξ) ≥ K · ξ2 for all ξ ∈ TM. The crucial point is that our lower curvature bounds are stable under an appropriate notion of Dconvergence of metric measure spaces. We define a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ≤ C is closed under Dconvergence. Moreover, the family of normalized metric measure spaces with doubling constant ≤ C and radius ≤ R is compact under Dconvergence.
The geometry of optimal transportation
 Acta Math
, 1996
"... A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measurepreserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of x − y. This map i ..."
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Cited by 254 (33 self)
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A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measurepreserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of x − y. This map is unique: it is characterized by the formula y(x) =x−(∇h) −1 (∇ψ(x)) and geometrical restrictions on ψ. Connections with mathematical economics, numerical computations, and the MongeAmpère equation are sketched. ∗ Both authors gratefully acknowledge the support provided by postdoctoral fellowships: WG at
Differential equations methods for the MongeKantorovich mass transfer problem
 Mem. Amer. Math. Soc
, 1999
"... We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfie ..."
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Cited by 170 (8 self)
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We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the pLaplacian equation −div(Dup  p−2 Dup) = f + − f − in the limit as p → ∞. The idea is to show up → u, where u satisfies Du  ≤ 1, −div(aDu) = f + − f − for some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du, f + and f −. Contents 1.
Continuity, curvature, and the general covariance of optimal transportation
"... Abstract. Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward t ..."
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Cited by 87 (17 self)
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Abstract. Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via nonnegativity of the sectional curvature of certain nullplanes in a novel but natural pseudoRiemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudoRiemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered. 1.
Constructive quantization: approximation by empirical measures
, 2012
"... In this article, we study the approximation of a probability measure µ on R d by its empirical measure ˆµN interpreted as a random quantization. As error criterion we consider an averaged pth moment Wasserstein metric. In the case where 2p < d, we establish fine upper and lower bounds for the er ..."
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Cited by 70 (2 self)
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In this article, we study the approximation of a probability measure µ on R d by its empirical measure ˆµN interpreted as a random quantization. As error criterion we consider an averaged pth moment Wasserstein metric. In the case where 2p < d, we establish fine upper and lower bounds for the error, a highresolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.
Perspectives of Monge Properties in Optimization
, 1995
"... An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) f ..."
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Cited by 70 (2 self)
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An m &times; n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS
, 2003
"... In transport problems of Monge’s types, the total cost of a transport map is usually an integral of some function of the distance, such as x − y  p. In many real applications, the actual cost may naturally be determined by a transport path. For shipping two items to one location, a “Y shaped” path ..."
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Cited by 58 (14 self)
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In transport problems of Monge’s types, the total cost of a transport map is usually an integral of some function of the distance, such as x − y  p. In many real applications, the actual cost may naturally be determined by a transport path. For shipping two items to one location, a “Y shaped” path may be preferable to a “V shaped ” path. Here, we show that any probability measure can be transported to another probability measure through a general optimal transport path, which is given by a vector measure in our setting. Moreover, we define a new distance on the space of probability measures which in fact metrizies the weak * topology of measures. Under this distance, the space of probability measures becomes a length space. Relations as well as related problems about transport paths and transport plans are also discussed in the end.
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 54 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
Some Applications of Mass Transport to GaussianType Inequalities
, 2002
"... As discovered by Brenier, mapping through a convex gradient gives the optimal transport in Rn. In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic ..."
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Cited by 50 (6 self)
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As discovered by Brenier, mapping through a convex gradient gives the optimal transport in Rn. In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.