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 PDF
 J. Amer. Math. Soc. 15 (2002), 126 Request permission
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 changeofvariables technique offering considerably more flexibility than existing approaches.References

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Additional Information
 Luis A. Caffarelli
 Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 787121082
 MR Author ID: 44175
 Email: caffarel@math.utexas.edu
 Mikhail Feldman
 Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
 MR Author ID: 226925
 Email: feldman@math.wisc.edu
 Robert J. McCann
 Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
 MR Author ID: 333976
 ORCID: 000000033867808X
 Email: mccann@math.toronto.edu
 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 21700699 RGPIN of the Natural Sciences and Engineering Research Council of Canada. The hospitality of the MaxPlanck 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), 126
 MSC (2000): Primary 49Q20; Secondary 26B10, 28A50, 58E17, 90B06
 DOI: https://doi.org/10.1090/S0894034701003769
 MathSciNet review: 1862796