Skip to main content
Log in

The geometry of optimal transportation

  • Published:
Acta Mathematica

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Abdellaoui, T., Détermination d'un couple optimal du problème de Monge-Kantorovitch.C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 981–984.

    MATH  MathSciNet  Google Scholar 

  2. Abdellaoui, T. &Heinich, H., Sur la distance de deux lois dans le cas vectoriel.C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 397–400.

    MathSciNet  Google Scholar 

  3. Alberti, G., On the structure of singular sets of convex functions.Calc. Var. Partial Differential Equations, 2 (1994), 17–27.

    Article  MATH  MathSciNet  Google Scholar 

  4. Aleksandrov, A. D., Existence and uniqueness of a convex surface with a given integral curvature.C. R. (Doklady) Acad. Sci. URSS (N.S.), 35 (1942), 131–134.

    MATH  MathSciNet  Google Scholar 

  5. Appell, P., Memoire sur les déblais et les remblais des systèmes continues ou discontinues.Mémoires présentés par divers Savants à l'Académie des Sciences de l'Institut de France, Paris, I. N., 29 (1887), 1–208.

    MathSciNet  Google Scholar 

  6. Balder, E. J., An extension of duality-stability relations to non-convex optimization problems.SIAM J. Control Optim. 15 (1977), 329–343.

    Article  MATH  MathSciNet  Google Scholar 

  7. Brenier, Y., Decomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805–808.

    MATH  MathSciNet  Google Scholar 

  8. —, Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44 (1991), 375–417.

    MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L., The regularity of mappings with a convex potential.J. Amer. Math. Soc., 5 (1992), 99–104.

    Article  MATH  MathSciNet  Google Scholar 

  10. —, Allocation maps with general cost functions, inPartial Differential Equations and Applications (P. Marcellini, G. Talenti and E. Vesintini, eds.), pp. 29–35. Lecture Notes in Pure and Appl. Math., 177. Dekker, New York, 1996.

    Google Scholar 

  11. Cuesta-Albertos, J. A. &Matrán, C., Notes on the Wasserstein metric in Hilbert spaces.Ann. Probab., 17 (1989), 1264–1276.

    MathSciNet  Google Scholar 

  12. Cuesta-Albertos, J. A., Matrán, C. & Tuero-Díaz, A., Properties of the optimal maps for theL 2-Monge-Kantorovich transportation problem. Preprint.

  13. Cuesta-Albertos, J. A., Rüschendorf, L. &Tuero-Díaz, A., Optimal coupling of multivariate distributions and stochastic processes.J. Multivariate Anal., 46 (1993), 335–361.

    Article  MathSciNet  Google Scholar 

  14. Cuesta-Albertos, J. A. &Tuero-Díaz, A., A characterization for the solution of the Monge-Kantorovich mass transference problem.Statist. Probab. Lett., 16 (1993), 147–152.

    Article  MathSciNet  Google Scholar 

  15. Dall'Aglio, G., Sugli estremi dei momenti delle funzioni di ripartizione-doppia.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 10 (1956), 35–74.

    MATH  MathSciNet  Google Scholar 

  16. Darboux, G., Prix Bordin (géométrie).C. R. Acad. Sci. Paris, 101 (1885), 1312–1316.

    Google Scholar 

  17. Douglis, A., Solutions in the large for multi-dimensional non-linear partial differential equations of first order.Ann. Inst. Fourier (Grenoble), 15:2 (1965), 1–35.

    MATH  MathSciNet  Google Scholar 

  18. Evans, L. C. & Gangbo, W., Differential equations methods for the Monge-Kantorovich mass transfer problem. Preprint.

  19. Fréchet, M., Sur la distance de deux lois de probabilité.C. R. Acad. Sci. Paris, 244 (1957), 689–692.

    MATH  MathSciNet  Google Scholar 

  20. Gangbo, W. &McCann, R. J., Optimal maps in Monge's mass transport problem.C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1653–1658.

    MathSciNet  Google Scholar 

  21. Kantorovich, L., On the translocation of masses.C. R. (Doklady) Acad. Sci. URSS (N. S.), 37 (1942), 199–201.

    MathSciNet  Google Scholar 

  22. —, On a problem of Monge.Uspekhi Mat. Nauk. 3 (1948), 225–226 (in Russian).

    Google Scholar 

  23. Katz, B. S. (editor),Nobel Laureates in Economic Sciences: A Biographical Dictionary. Garland Publishing Inc., New York, 1989.

    Google Scholar 

  24. Kellerer, H. G., Duality theorems for marginal problems.Z. Wahrsch. Verw. Gebiete, 67 (1984), 399–432.

    Article  MATH  MathSciNet  Google Scholar 

  25. Knott, M. &Smith, C. S., On the optimal mapping of distributions.J. Optim. Theory Appl., 43 (1984), 39–49.

    Article  MathSciNet  Google Scholar 

  26. Levin, V. L., General Monge-Kantorovich problem and its applications in measure theory and mathematical economics, inFunctional Analysis, Optimization, and Mathematical Economics (L. J. Leifman, ed.), pp. 141–176. Oxford Univ. Press, New York, 1990.

    Google Scholar 

  27. McCann, R. J., Existence and uniqueness of monotone measure-preserving maps.Duke Math. J., 80 (1995), 309–323.

    Article  MATH  MathSciNet  Google Scholar 

  28. McCann, R. J., A convexity principle for interacting gases. To appear inAdv. in Math.

  29. McCann, R. J., Exact solutions to the transportation problem on the line. To appear.

  30. Monge, G., 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, 1781, pp. 666–704.

  31. Rachev, S. T., The Monge-Kantorovich mass transference problem and its stochastic applications.Theory Probab. Appl., 29 (1984), 647–676.

    Article  Google Scholar 

  32. Rockafellar, R. T., Characterization of the subdifferentials of convex functions.Pacific J. Math., 17 (1966), 497–510.

    MATH  MathSciNet  Google Scholar 

  33. —,Convex Analysis. Princeton Univ. Press, Princeton, NJ, 1972.

    Google Scholar 

  34. Rüschendorf, L., Bounds for distributions with multivariate marginals, inStochastic Orders and Decision Under Risk (K. Mosler and M. Scarsini, eds.), pp. 285–310. IMS Lecture Notes-Monograph Series. Inst. of Math. Statist, Hayward, CA, 1991.

    Google Scholar 

  35. —, Frechet bounds and their applications, inAdvances in Probability Distributions with Given Marginals (G. Dall'Aglio et al., eds.), pp. 151–187. Math. Appl., 67. Kluwer Acad. Publ., Dordrecht, 1991.

    Google Scholar 

  36. —, Optimal solutions of multivariate coupling problems.Appl. Math. (Warszaw), 23 (1995), 325–338.

    MATH  Google Scholar 

  37. —, Onc-optimal random variables.Statist. Probab. Lett., 27 (1996), 267–270.

    Article  MATH  MathSciNet  Google Scholar 

  38. Rüschendorf, L. &Rachev, S. T., A characterization of random variables with minimumL 2-distance.J. Multivariate Anal., 32 (1990), 48–54.

    Article  MathSciNet  Google Scholar 

  39. Schneider, R.,Convex Bodies: The Brunn-Minkowski Theory. Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

  40. Smith, C. &Knott, M., On the optimal transportation of distributions.J. Optim. Theory Appl., 52 (1987), 323–329.

    Article  MathSciNet  Google Scholar 

  41. —, On Hoeffding-Fréchet bounds and cyclic monotone relations.J. Multivariate Anal., 40 (1992), 328–334.

    Article  MathSciNet  Google Scholar 

  42. Sudakov, V. N., Geometric problems in the theory of infinite-dimensional probability distributions.Proc. Steklov Inst. Math., 141 (1979), 1–178.

    MathSciNet  Google Scholar 

  43. Zajíĉek, L., On the differentiation of convex functions in finite and infinite dimensional spaces.Czechoslovak Math. J. 29 (104) (1979), 340–348.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gangbo, W., McCann, R.J. The geometry of optimal transportation. Acta Math. 177, 113–161 (1996). https://doi.org/10.1007/BF02392620

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02392620

Keywords

Navigation