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Monge's transport problem on a Riemannian manifold


Authors: Mikhail Feldman and Robert J. McCann
Journal: Trans. Amer. Math. Soc. 354 (2002), 1667-1697
MSC (2000): Primary 49Q20, 28A50
DOI: https://doi.org/10.1090/S0002-9947-01-02930-0
Published electronically: December 4, 2001
MathSciNet review: 1873023
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Abstract: Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures $\mu^+ \ne \mu^-$, find the measure-preserving map $s:M \longrightarrow M$ between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold $M$ -- and assuming absolute continuity of $\mu^+$ -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.


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Additional Information

Mikhail Feldman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: feldman@math.wisc.edu

Robert J. McCann
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: mccann@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02930-0
Keywords: Monge-Kantorovich mass transportation, Riemannian manifold, optimal map, dual problem
Received by editor(s): March 30, 2001
Published electronically: December 4, 2001
Additional Notes: The authors gratefully acknowledge the support of grants DMS 0096090 [MF] and 0074037 [MF and RJM] of the U.S. National Science Foundation, and grant 217006-99 [RJM] of the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2001 by the authors

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