## 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

## 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.## References

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## Additional Information

**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: 0000-0003-3867-808X
- Email: mccann@math.toronto.edu
- 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: https://doi.org/10.1090/S0002-9947-01-02930-0
- MathSciNet review: 1873023