Advertisement

Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 377–389 | Cite as

On the Equality of Values in the Monge and Kantorovich Problems

  • V. I. BogachevEmail author
  • A. N. Kalinin
  • S. N. Popova
Article
  • 4 Downloads

This paper is concerned with the study of conditions under which the Monge and Kantorovich problems with a continuous cost function on a product of two completely regular spaces and two given atomless Radon measures-projections on these spaces have equal values of the corresponding infima.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. I. Bogachev, Weak Convergence of Measures [in Russian], Institute for Computer Research, Moscow–Izhevsk (2016).Google Scholar
  2. 2.
    V. I. Bogachev and A. N. Kalinin, “A continuous cost function for which the minima in the Monge and Kantorovich problems are not equal,” Dokl. Akad. Nauk, 463, 383–386 (2015).MathSciNetzbMATHGoogle Scholar
  3. 3.
    V. I. Bogachev and A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives,” Usp. Mat. Nauk, 67, 3–110 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, “Triangular transformations of measures,” Matem. Sb., 196, 3–30 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A.M. Vershik, “Some remarks on the infinite-dimensional problems of linear programming,” Usp Mat. Nauk, 25, 117–124 (1970).MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. M. Vershik, “The Kantorovich metric: the initial history and little-known applications,” Zap. Nauchn. Semin. POMI, 312, 69–85 (2004).Google Scholar
  7. 7.
    A. M. Vershik, P. B. Zatitskii and F. V. Petrov, “Virtual continuity of measurable functions and its applications,” Usp Mat. Nauk, 69, 81–114 (2014).MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. V. Kantorovich, Mathematical Methods of Organizing and Planning Production [in Russian], Leningr. State Univ., Leningrad (1939).Google Scholar
  9. 9.
    L. V. Kantorovitch, “On the translocation of masses,” Dokl. Akad. Nauk SSSR, 37, 227–229 (1942).MathSciNetzbMATHGoogle Scholar
  10. 10.
    L. V. Kantorovich, “On a problem of Monge,” Usp. Mat. Nauk, 3, 225–226 (1948).Google Scholar
  11. 11.
    L. V. Kantorovich and G. Sh. Rubinshtein, “On a functional space and certain extremum problems,” Dokl. Akad. Nauk SSSR, 115, 1058–1061 (1957).MathSciNetzbMATHGoogle Scholar
  12. 12.
    L. V. Kantorovich and G. Sh. Rubinshtein, “On a space of completely additive functions,” Vestnik Leningr. Univ., 7, No. 2, 52–59 (1958).MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. A. Lipchius, “A note on the equality in the problems of Monge and Kantorovich,” Teor. Verojatn. Primen., 50, 779–782 (2005)CrossRefGoogle Scholar
  14. 14.
    V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions,” Trudy Mat. Inst. Steklov, 141, 1–190 (1976).MathSciNetGoogle Scholar
  15. 15.
    L. Ambrosio, “Lecture notes on optimal transport problems,” Lect. Notes Math., 1812, 1–52 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Ambrosio and N. Gigli, “A user’s guide to optimal transport,” Lect. Notes Math., 2062, 1–155 (2013).MathSciNetCrossRefGoogle Scholar
  17. 17.
    L. Ambrosio, B. Kirchheim, and A. Pratelli, “Existence of optimal transport maps for chrystalline norms,” Duke Math. J., 125, 207–241 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    L. Ambrosio and A. Pratelli, “Existence and stability results in the L 1 theory of optimal transportation,” In: Optimal Transportation and Applications (Martina Franca, 2001), Lect Notes Math., 1813, (2003), pp. 123–160.Google Scholar
  19. 19.
    P. Appel, “Mèmoire sur les dèblais et les remblais des systèmes continus ou discontinus,” M´emoires prèsentès par divers Savants à l’Acadèmie des Sciences de l’Institut de France, Paris, 29, 1–208 (1887).Google Scholar
  20. 20.
    M. Beiglböck, M. Goldstern, G. Maresch, and W. Schachermayer, “Optimal and better transport plans,” J. Funct. Anal., 256, 1907–1927 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Beiglböck and W. Schachermayer, “Duality for Borel measurable cost functions,” Trans. Amer. Math. Soc., 363, 4203–4224 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. Bianchini and F. Cavalletti, “The Monge problem for distance cost in geodesic spaces,” Comm. Math. Phys., 318, 615–673 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    S. Bianchini and S. Daneri, “On Sudakov’s type decomposition of transference plans with norm costs,” Mem. Amer. Math. Soc. (to appear); arXiv 1311.1918v2.Google Scholar
  24. 24.
    V. I. Bogachev, Measure Theory, Vol. 2, Springer, Berlin (2007).Google Scholar
  25. 25.
    L. A. Caffarelli, M. Feldman, and R. J. McCann, “Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs,” J. Amer. Math. Soc., 15, 1–26 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    L. Caravenna, “A proof of Sudakov theorem with strictly convex norms,” Math. Z., 268, 371–407 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    L. Caravenna, “A proof of Monge problem in ℝn by stability,” Rend. Inst. Mat. Univ. Trieste, 43, 31–51 (2011).MathSciNetzbMATHGoogle Scholar
  28. 28.
    F. Cavalletti, “Monge problem in metric measure spaces with Riemannian curvaturedimension condition,” Nonlinear Anal., 99, 136–151 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    T. Champion and L. De Pascale, “The Monge problem for strictly convex norms in ℝd,” J. Eur. Math. Soc., 12, 1355–1369 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    T. Champion and L. De Pascale, “The Monge problem in ℝd,” Duke Math. J., 157, 551–572 (2011).Google Scholar
  31. 31.
    T. Champion and L. De Pascale, “The Monge problem in ℝd: variations on a theme,” Zap. Nauchn. Semin. POMI, 390, 182–200 (2011).Google Scholar
  32. 32.
    G. A. Edgar, “Measurable weak sections,” Illinois J. Math., 20, 630–646 (1976).Google Scholar
  33. 33.
    L. C. Evans and W. Gangbo, “Differential equations methods for the Monge–Kantorovich mass transfer problem,” Mem. Amer. Math. Soc., 137, No. 653, viii+66 p. (1999).Google Scholar
  34. 34.
    M. Feldman and R. McCann, “Monge’s transport problem on a Riemannian manifold,” Trans. Amer. Math. Soc., 354, 1667–1697 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    D. H. Fremlin, “Measurable functions and almost continuous functions,” Manuscr. Math., 33, 387–405 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    D. Fremlin, Measure Theory, Vols. 1–5. Univ. of Essex, Colchester (2000–2003).Google Scholar
  37. 37.
    G. Monge, “Mémoire sur la théorie des déblais et de remblais,” Histoire de l’Académie Royale des Sciences de Paris, 666–704 (1781).Google Scholar
  38. 38.
    A. Pratelli, “On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation,” Ann. Inst. H. Poincaré (B) Probab. Statist., 43, 1–13 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Vols. 1 and 2, Springer, New York (1998).Google Scholar
  40. 40.
    N. S. Trudinger and X.-J. Wang, “On the Monge mass transfer problem,” Calc. Var. Partial Differ. Equ., 13, 19–31 (2001).Google Scholar
  41. 41.
    C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence, Rhode Island(2003).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. I. Bogachev
    • 1
    Email author
  • A. N. Kalinin
    • 2
  • S. N. Popova
    • 2
  1. 1.Lomonosov Moscow State University; National Research University Higher School of Economics; Orthodox St.Tikhon Humanitarian UniversityMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations