On the equality of values in the Monge and Kantorovich problems

V. I. Bogachev   ; A. N. Kalinin   ; S. N. Popova

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 53-73

Résumé

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

Bibliographie

[1] V. I. Bogachev, Slabaya skhodimost mer, Institut kompyuternykh issledovanii, M.–Izhevsk, 2016

[2] V. I. Bogachev, A. N. Kalinin, “Nepreryvnaya funktsiya stoimosti, dlya kotoroi minimumy v zadachakh Monzha i Kantorovicha ne ravny”, Dokl. RAN, 463:4 (2015), 383–386 | DOI | Zbl

[3] V. I. Bogachev, A. V. Kolesnikov, “Zadacha Monzha–Kantorovicha: dostizheniya, svyazi i perspektivy”, Uspekhi matem. nauk, 67:5 (2012), 3–110 | DOI | MR | Zbl

[4] V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev, “Treugolnye preobrazovaniya mer”, Matem. sb., 196:3 (2005), 3–30 | DOI | MR | Zbl

[5] A. M. Vershik, “Neskolko zamechanii o beskonechnomernykh zadachakh lineinogo programmirovaniya”, Uspekhi matem. nauk, 25:5 (1970), 117–124 | MR | Zbl

[6] A. M. Vershik, “Metrika Kantorovicha: nachalnaya istoriya i maloizvestnye primeneniya”, Zap. nauchn. sem. POMI, 312, 2004, 69–85 | MR | Zbl

[7] A. M. Vershik, P. B. Zatitskii, F. V. Petrov, “Virtualnaya nepreryvnost izmerimykh funktsii mnogikh peremennykh i ee prilozheniya”, Uspekhi matem. nauk, 69:6 (2014), 81–114 | DOI | MR | Zbl

[8] L. V. Kantorovich, Matematicheskie metody organizatsii i planirovaniya proizvodstva, Izd-vo LGU, L., 1939; Изд. дом СПбГУ, СПб., 2012

[9] L. V. Kantorovich, “O peremeschenii mass”, Dokl. AN SSSR, 37:7–8 (1942), 227–229

[10] L. V. Kantorovich, “Ob odnoi probleme Monzha”, Uspekhi matem. nauk, 3:2 (1948), 225–226

[11] L. V. Kantorovich, G. Sh. Rubinshtein, “Ob odnom funktsionalnom prostranstve i nekotorykh ekstremalnykh zadachakh”, Dokl. AN SSSR, 115:6 (1957), 1058–1061 | MR | Zbl

[12] L. V. Kantorovich, G. Sh. Rubinshtein, “Ob odnom prostranstve vpolne additivnykh funktsii mnozhestva”, Vestn. LGU, 7:2 (1958), 52–59 | Zbl

[13] A. A. Lipchyus, “Zamechanie o ravenstve v zadachakh Monzha i Kantorovicha”, Teoriya veroyatn. i ee primen., 50:4 (2005), 779–782 | DOI | MR | Zbl

[14] V. N. Sudakov, “Geometricheskie problemy teorii beskonechnomernykh veroyatnostnykh raspredelenii”, Tr. Mat. in-ta AN SSSR, 141, 1976, 3–191 | MR | Zbl

[15] L. Ambrosio, “Lecture notes on optimal transport problems”, Lecture Notes in Math., 1812, 2003, 1–52 | DOI | MR | Zbl

[16] L. Ambrosio, N. Gigli, “A user's guide to optimal transport”, Lecture Notes in Math., 2062, 2013, 1–155 | DOI | MR

[17] L. Ambrosio, B. Kirchheim, A. Pratelli, “Existence of optimal transport maps for chrystalline norms”, Duke Math. J., 125:2 (2004), 207–241 | DOI | MR | Zbl

[18] L. Ambrosio, A. Pratelli, “Existence and stability results in the $L^1$ theory of optimal transportation”, Optimal transportation and applications (Martina Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, 2003, 123–160 | DOI | MR | Zbl

[19] P. Appel, “Mémoire sur les déblais et les remblais des systèmes continus ou discontinus”, Mémoires présentés par divers Savants à l'Académie des Sciences de l'Institut de France (Paris), 29 (1887), 1–208

[20] M. Beiglböck, M. Goldstern, G. Maresch, W. Schachermayer, “Optimal and better transport plans”, J. Funct. Anal., 256:6 (2009), 1907–1927 | DOI | MR | Zbl

[21] M. Beiglböck, W. Schachermayer, “Duality for Borel measurable cost functions”, Trans. Amer. Math. Soc., 363:8 (2011), 4203–4224 | DOI | MR | Zbl

[22] S. Bianchini, F. Cavalletti, “The Monge problem for distance cost in geodesic spaces”, Comm. Math. Phys., 318 (2013), 615–673 | DOI | MR | Zbl

[23] S. Bianchini, S. Daneri, “On Sudakov's type decomposition of transference plans with norm costs”, Mem. Amer. Math. Soc. (to appear) | MR

[24] V. I. Bogachev, Measure Theory, v. 2, Springer, Berlin, 2007 | MR | Zbl

[25] L. A. Caffarelli, M. Feldman, R. J. McCann, “Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs”, J. Amer. Math. Soc., 15:1 (2002), 1–26 | DOI | MR | Zbl

[26] L. Caravenna, “A proof of Sudakov theorem with strictly convex norms”, Math. Z., 268:1–2 (2011), 371–407 | DOI | MR | Zbl

[27] L. Caravenna, “A proof of Monge problem in $\mathbb R^n$ by stability”, Rend. Istit. Mat. Univ. Trieste, 43 (2011), 31–51 | MR | Zbl

[28] F. Cavalletti, “Monge problem in metric measure spaces with Riemannian curvature-dimension condition”, Nonlinear Anal., 99 (2014), 136–151 | DOI | MR | Zbl

[29] T. Champion, L. De Pascale, “The Monge problem for strictly convex norms in $\mathbb R^d$”, J. Eur. Math. Soc., 12:6 (2010), 1355–1369 | DOI | MR | Zbl

[30] T. Champion, L. De Pascale, “The Monge problem in $\mathbb R^d$”, Duke Math. J., 157:3 (2011), 551–572 | DOI | MR | Zbl

[31] J. Math. Sci. (New York), 181:6 (2012), 856–866 | DOI | MR | Zbl

[32] G. A. Edgar, “Measurable weak sections”, Illinois J. Math., 20:4 (1976), 630–646 | MR | Zbl

[33] L. C. Evans, W. Gangbo, Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137, no. 653, 1999, viii+66 pp. | MR

[34] M. Feldman, R. McCann, “Monge's transport problem on a Riemannian manifold”, Trans. Amer. Math. Soc., 354 (2002), 1667–1697 | DOI | MR | Zbl

[35] D. H. Fremlin, “Measurable functions and almost continuous functions”, Manuscr. Math., 33:3–4 (1981), 387–405 | DOI | MR | Zbl

[36] D. Fremlin, Measure Theory, v. 1–5, University of Essex, Colchester, 2000–2003 | MR

[37] G. Monge, “Mémoire sur la théorie des déblais et de remblais”, Histoire de l'Académie Royale des sciences de Paris, 1781, 666–704

[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 (2007), 1–13 | DOI | MR | Zbl

[39] S. T. Rachev, L. Rüschendorf, Mass Transportation Problems, v. I, II, Springer, New York, 1998 | MR | Zbl

[40] N. S. Trudinger, X.-J. Wang, “On the Monge mass transfer problem”, Calc. Var. Partial Differ. Equ., 13:1 (2001), 19–31 | DOI | MR | Zbl

[41] C. Villani, Topics in Optimal Transportation, Amer. Math. Soc., Providence, Rhode Island, 2003 | MR | Zbl