Geodesic
The Monge problem in $\mathbb R^d$: Variations on a theme
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 182-200 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a recent paper the authors proved that, under natural assumptions on the first marginal, the Monge problem in $\mathbb R^d$ for cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of uniformly convex norm and we will also use very recent results by other authors [1]. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set given in the particular case of this paper is original. 
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Thierry Champion; Luigi De Pascale. The Monge problem in $\mathbb R^d$: Variations on a theme. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 182-200. http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a6/

[1] N. Ahmad, H. K. Kim, R. McCann, Extremal doubly stochastic measures and optimal transportation, Preprint, temporarily available on http://www.math.toronto.edu/mccann/

[2] L. Ambrosio, “Lecture Notes on Optimal Transportation Problems”, Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes Math., 1812, Springer, Berlin, 2003, 1–52 | DOI | MR | Zbl

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

[4] 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 Math., 1813, Springer, Berlin, 123–160 | DOI | MR | Zbl

[5] G. Anzellotti, S. Baldo, “Asymptotic development by $\Gamma$-convergence”, Appl. Math. Optim., 27:2 (1993), 105–123 | DOI | MR | Zbl

[6] H. Attouch, “Viscosity solutions of minimization problems”, SIAM J. Optim., 6:3 (1996), 769–806 | DOI | MR | Zbl

[7] 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

[8] L. Caravenna, “A partial proof of Sudakov theorem via disintegration of measures”, Mathematische Zeitschrift (to appear); Preprint, SISSA, 2008 available at http://cvgmt.sns.it

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

[10] T. Champion, L. De Pascale, “The Monge problem in $\mathbb R^d$”, Duke Math. J. (to appear) temporarily available on http://cvgmt.sns.it

[11] T. Champion, L. De Pascale, P. Juutinen, “The $\infty$-Wasserstein distance: local solutions and existence of optimal transport maps”, SIAM J. of Math. Anal., 40:1 (2008), 1–20 | DOI | MR | Zbl

[12] L. C. Evans, W. Gangbo, Differential Equations Methods for the Monge–Kantorovich Mass Transfer Problem, Mem. Amer. Math. Soc., 137, 1999 | MR

[13] L. V. Kantorovich, “On the translocation of masses”, C. R. Dokl. Acad. Sci. URSS, 37 (1942), 199–201 | MR

[14] L. V. Kantorovich, “On a problem of Monge”, Uspekhi Mat. Nauk, 3:2 (1948), 225–226 (in Russian)

[15] G. Monge, Mémoire sur la théorie des Déblais et des Remblais, Histoire de l'Académie des Sciences de Paris, 1781

[16] A. Pratelli, “On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation”, Ann. Inst. H. Poincaré Probab. Statist., 43:1 (2007), 1–13 | DOI | MR | Zbl

[17] A. Pratelli, “On the sufficiency of c-cyclical monotonicity for optimality of transport plans”, Math. Z., 258:3 (2008), 677–690 | DOI | MR | Zbl

[18] F. Santambrogio, “Absolute continuity and summability of optimal transport densities: simpler proofs and new estimates”, Calc. Var. Partial Differential Equations, 36:3 (2009), 343–354 | DOI | MR | Zbl

[19] V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Cover to cover translation of Trudy Mat. Inst. Steklov, 141 (1976), Proc. Steklov Inst. Math., 141, 1979, 178 pp. | MR | MR | Zbl

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

[21] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003 | MR | Zbl

[22] C. Villani, Optimal Transport. Old and New, Grundlehren der mathematischen Wissenschaften, 338, Springer, Berlin–Heidelberg, 2008 | MR