The geometry of optimal transportation (original) (raw)

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, in_Partial 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 the_L_ 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, in_Functional 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 in_Adv. 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, in_Stochastic 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, in_Advances 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. —, On_c_-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 minimum_L_ 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

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