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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs
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by Luis A. Caffarelli, Mikhail Feldman and Robert J. McCann
J. Amer. Math. Soc. 15 (2002), 1-26
Published electronically: July 31, 2001


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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Bibliographic Information
  • Luis A. Caffarelli
  • Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
  • MR Author ID: 44175
  • Email:
  • Mikhail Feldman
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
  • MR Author ID: 226925
  • Email:
  • Robert J. McCann
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
  • MR Author ID: 333976
  • ORCID: 0000-0003-3867-808X
  • Email:
  • 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.
  • © Copyright 2001 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 15 (2002), 1-26
  • MSC (2000): Primary 49Q20; Secondary 26B10, 28A50, 58E17, 90B06
  • DOI:
  • MathSciNet review: 1862796