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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Monge’s transport problem on a Riemannian manifold
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by Mikhail Feldman and Robert J. McCann PDF
Trans. Amer. Math. Soc. 354 (2002), 1667-1697


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
  • 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 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.
  • © Copyright 2001 by the authors
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1667-1697
  • MSC (2000): Primary 49Q20, 28A50
  • DOI:
  • MathSciNet review: 1873023