Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

 

 

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs


Authors: Luis A. Caffarelli, Mikhail Feldman and Robert J. McCann
Journal: J. Amer. Math. Soc. 15 (2002), 1-26
MSC (2000): Primary 49Q20; Secondary 26B10, 28A50, 58E17, 90B06
Published electronically: July 31, 2001
MathSciNet review: 1862796
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 change-of-variables technique offering considerably more flexibility than existing approaches.


References [Enhancements On Off] (What's this?)

  • 1. G. Alberti, B. Kircheim and D. Preiss. Presented in a lecture by Kircheim at the Scuola Normale Superiori workshop, October 27, 2000. See also [2, Remark 6.1].
  • 2. L. Ambrosio. Lecture notes on optimal transport problems. To appear with Proceedings of a Centro Internazionale Matematico Estivo Summer School in the Springer-Verlag Lecture Notes in Mathematics Series.
  • 3. Keith Ball, Eric A. Carlen, and Elliott H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), no. 3, 463–482. MR 1262940, 10.1007/BF01231769
  • 4. Luis A. Caffarelli, Allocation maps with general cost functions, Partial differential equations and applications, Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996, pp. 29–35. MR 1371577
  • 5. Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 65–126. MR 1698853
  • 6. Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • 7. L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, 10.1090/memo/0653
  • 8. Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
  • 9. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • 10. Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. MR 0110078, 10.1090/S0002-9947-1959-0110078-1
  • 11. Mikhail Feldman, Variational evolution problems and nonlocal geometric motion, Arch. Ration. Mech. Anal. 146 (1999), no. 3, 221–274. MR 1720391, 10.1007/s002050050142
  • 12. M. Feldman, R.J. McCann. Uniqueness and transport density in Monge's mass transportation problem. To appear in Calc. Var. Partial Differential Equations.
  • 13. M. Feldman, R.J. McCann. Monge's transport problem on a Riemannian manifold. Submitted to Trans. Amer. Math. Soc.
  • 14. Wilfrid Gangbo and Robert J. McCann, Optimal maps in Monge’s mass transport problem, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 12, 1653–1658 (English, with English and French summaries). MR 1367824
  • 15. Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161. MR 1440931, 10.1007/BF02392620
  • 16. Paul R. Halmos, The decomposition of measures, Duke Math. J. 8 (1941), 386–392. MR 0004718
  • 17. L. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199-201, 1942.
  • 18. R.J. McCann. Polar factorization of maps on Riemannian manifolds. To appear in Geom. Funct. Anal.
  • 19. G. Monge. Mémoire sur la théorie des déblais et de remblais. Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666-704, 1781.
  • 20. Svetlozar T. Rachev and Ludger Rüschendorf, Mass transportation problems. Vol. I, Probability and its Applications (New York), Springer-Verlag, New York, 1998. Theory. MR 1619170
    Svetlozar T. Rachev and Ludger Rüschendorf, Mass transportation problems. Vol. II, Probability and its Applications (New York), Springer-Verlag, New York, 1998. Applications. MR 1619171
  • 21. V.A. Rokhlin. On the fundamental ideas of measure theory. Mat. Sbornik (N.S.), 25(67):107-150, 1949.
  • 22. Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
  • 23. V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math. 2 (1979), i–v, 1–178. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976). MR 530375
  • 24. N.S. Trudinger, X.-J. Wang. On the Monge mass transfer problem. To appear in Calc. Var. Partial Differential Equations.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 49Q20, 26B10, 28A50, 58E17, 90B06

Retrieve articles in all journals with MSC (2000): 49Q20, 26B10, 28A50, 58E17, 90B06


Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
Email: caffarel@math.utexas.edu

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: http://dx.doi.org/10.1090/S0894-0347-01-00376-9
Keywords: Monge-Kantorovich mass transportation, resource allocation, optimal map, optimal coupling, infinite dimensional linear programming, dual problem, Wasserstein distance
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.
Article copyright: © Copyright 2001 American Mathematical Society