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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.
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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Cited by 4 (2 self)
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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
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
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You can submit your corrections online, via email or by fax. For online submission please insert your corrections in the online correction form. Always indicate the line number to which the correction refers. You can also insert your corrections in the proof PDF and email the annotated PDF. For fax submission, please ensure that your corrections are clearly legible. Use a fine black pen and write the correction in the margin, not too close to the edge of the page. Remember to note the journal title, article number, and your name when sending your response via email or fax. Check the metadata sheet to make sure that the header information, especially author names and the corresponding affiliations are correctly shown. Check the questions that may have arisen during copy editing and insert your answers/ corrections. Check that the text is complete and that all figures, tables and their legends are included. Also check the accuracy of special characters, equations, and electronic supplementary material if applicable. If necessary refer to the Edited manuscript. The publication of inaccurate data such as dosages and units can have serious consequences. Please take particular care that all such details are correct. Please do not make changes that involve only matters of style. We have generally introduced forms that follow the journal’s style. Substantial changes in content, e.g., new results, corrected values, title and authorship are not allowed without the approval of the responsible editor. In such a case, please contact the Editorial Office and return his/her consent together with the proof. If we do not receive your corrections within 48 hours, we will send you a reminder. Your article will be published Online First approximately one week after receipt of your corrected proofs. This is the official first publication citable with the DOI. Further changes are, therefore, not possible. The printed version will follow in a forthcoming issue. Please note After online publication, subscribers (personal/institutional) to this journal will have access to the complete article via the DOI using the
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
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