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Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs

Authors: Luis A. Caffarelli, Mikhail Feldman and Robert J. McCann
Journal: J. Amer. Math. Soc. 15 (2002), 1-26
MSC (2000): Primary 49Q20; Secondary 26B10, 28A50, 58E17, 90B06
Published electronically: July 31, 2001
MathSciNet review: 1862796
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Abstract | References | Similar Articles | Additional Information

Abstract: Given two densities on $\mathbf {R}^n$ with the same total mass, the Monge transport problem is to find a Borel map $s:\mathbf {R}^n \to \mathbf {R}^n$ rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.

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Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
MR Author ID: 44175

Mikhail Feldman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
MR Author ID: 226925

Robert J. McCann
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
MR Author ID: 333976
ORCID: 0000-0003-3867-808X

Keywords: Monge-Kantorovich mass transportation, resource allocation, optimal map, optimal coupling, infinite dimensional linear programming, dual problem, Wasserstein distance
Received by editor(s): March 15, 2000
Published electronically: July 31, 2001
Additional Notes: This research was supported by grants DMS 9714758, 9623276, 9970577, and 9622997 of the US National Science Foundation, and grant 217006-99 RGPIN of the Natural Sciences and Engineering Research Council of Canada. The hospitality of the Max-Planck Institutes at Bonn and Leipzig are gratefully acknowledged by the second and third authors respectively.
Article copyright: © Copyright 2001 American Mathematical Society