# American Institute of Mathematical Sciences

April  2014, 34(4): 1339-1353. doi: 10.3934/dcds.2014.34.1339

## On the twist condition and $c$-monotone transport plans

 1 Université de Toulon, IMATH, EA 2134, 83957 La Garde, France 2 Dipartimento di Matematica Applicata, Università di Pisa, Via Filippo Buonarroti 1/c, 56127 Pisa, Italy

Received  October 2012 Revised  March 2013 Published  October 2013

A usual approach for proving the existence of an optimal transport map, be it in ${\mathbb R}^d$ or on more general manifolds, involves a regularity condition on the transport cost (the so-called Left Twist condition, i.e. the invertibility of the gradient in the first variable) as well as the fact that any optimal transport plan is supported on a cyclically-monotone set. Under the classical assumption that the initial measure does not give mass to sets with $\sigma$-finite $\mathcal{H}^{d-1}$ measure and a stronger regularity condition on the cost (the Strong Left Twist), we provide a short and self-contained proof of the fact that any feasible transport plan (optimal or not) satisfying a $c$-monotonicity assumption is induced by a transport map. We also show that the usual costs induced by Tonelli Lagrangians satisfy the Strong Left Twist condition we propose.
Citation: Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339
##### References:
 [1] Luigi Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [2] Luigi Ambrosio and Aldo Pratelli, Existence and stability results in the $L^1$ theory of optimal transportation, in "Optimal Transportation and Applications" (Martina Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, (2003), 123-160. doi: 10.1007/978-3-540-44857-0_5. [3] Patrick Bernard and Boris Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74. [4] Yann Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 805-808. [5] Luis A. Caffarelli, Allocation maps with general cost functions, in "Partial Differential Equations and Applications," Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, (1996), 29-35. [6] Guillaume Carlier, Duality and existence for a class of mass transportation problems and economic applications, in "Advances in Mathematical Economics. Vol. 5," Adv. Math. Econ., 5, Springer, Tokyo, (2003), 1-21. doi: 10.1007/978-4-431-53979-7_1. [7] Laura Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Z., 268 (2011), 371-407. doi: 10.1007/s00209-010-0677-6. [8] Thierry Champion and Luigi De Pascale, The Monge problem for strictly convex norms in $\mathbbR^d$, J. Eur. Math. Soc. (JEMS), 12 (2010), 1355-1369. doi: 10.4171/JEMS/234. [9] ______, The Monge problem in $\mathbbR^d$, Duke Math. J., 157 (2011), 551-572. doi: 10.1215/00127094-1272939. [10] ______, The Monge problem in $\mathbbR^d$: Variations on a theme, Journal of Mathematical Sciences, 181 (2012), 856-866. doi: 10.1007/s10958-012-0719-1. [11] Thierry Champion, Luigi De Pascale and Petri Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20. doi: 10.1137/07069938X. [12] Dario Cordero-Erausquin, Sur le transport de mesures périodiques, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 199-202. doi: 10.1016/S0764-4442(00)88593-6. [13] Luis A. Caffarelli, Mikhail Feldman and Robert J. McCann, Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), 1-26 (electronic). doi: 10.1090/S0894-0347-01-00376-9. [14] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999). doi: 10.1090/memo/0653. [15] Albert Fathi and Alessio Figalli, Optimal transportation on non-compact manifolds, Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5. [16] Alessio Figalli, Existence, uniqueness, and regularity of optimal transport maps, SIAM J. Math. Anal., 39 (2007), 126-137. doi: 10.1137/060665555. [17] _______, The Monge problem on non-compact manifolds, Rend. Semin. Mat. Univ. Padova, 117 (2007), 147-166. [18] Wilfrid Gangbo and Robert J. McCann, Optimal maps in Monge's mass transport problem, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1653-1658. [19] _______, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620. [20] Nassif Ghoussoub and Abbas Moameni, A self-dual polar factorization for vector fields, Comm. Pure Appl. Math., 66 (2013), 905-933. doi: 10.1002/cpa.21430. [21] L. V. Kantorovith, On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS (N. S.), 37 (1942), 199-201. [22] _______, On a problem of Monge, (in Russian) Uspekhi Mat. Nauk., 3 (1948), 225-226. [23] Eckehard Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Analysis, 9 (1985), 1381-1399. doi: 10.1016/0362-546X(85)90097-5. [24] Vladimir Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-Valued Anal., 7 (1999), 7-32. doi: 10.1023/A:1008753021652. [25] Robert J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679. [26] Gaspard Monge, "Mémoire sur la Théorie des Déblais et des Remblais," Histoire de l'Académie des Sciences de Paris, (1781). [27] A. Pratelli, On the sufficiency of $c$-cyclical monotonicity for optimality of transport plans, Math. Z., 258 (2008), 677-690. doi: 10.1007/s00209-007-0191-7. [28] L. Rüschendorf and S. T. Rachev, A characterization of random variables with minimum $L^2$-distance, J. Multivariate Anal., 32 (1990), 48-54. doi: 10.1016/0047-259X(90)90070-X. [29] Ludger Rüschendorf, On $c$-optimal random variables, Statist. Probab. Lett., 27 (1996), 267-270. doi: 10.1016/0167-7152(95)00078-X. [30] Walter Schachermayer and Josef Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proc. Amer. Math. Soc., 137 (2009), 519-529. doi: 10.1090/S0002-9939-08-09419-7. [31] Neil S. Trudinger and Xu-Jia Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations, 13 (2001), 19-31. doi: 10.1007/PL00009922. [32] Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. [33] _______, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

##### References:
 [1] Luigi Ambrosio, Lecture notes on optimal transport problems, in "Mathematical Aspects of Evolving Interfaces" (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [2] Luigi Ambrosio and Aldo Pratelli, Existence and stability results in the $L^1$ theory of optimal transportation, in "Optimal Transportation and Applications" (Martina Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, (2003), 123-160. doi: 10.1007/978-3-540-44857-0_5. [3] Patrick Bernard and Boris Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121. doi: 10.4171/JEMS/74. [4] Yann Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 805-808. [5] Luis A. Caffarelli, Allocation maps with general cost functions, in "Partial Differential Equations and Applications," Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, (1996), 29-35. [6] Guillaume Carlier, Duality and existence for a class of mass transportation problems and economic applications, in "Advances in Mathematical Economics. Vol. 5," Adv. Math. Econ., 5, Springer, Tokyo, (2003), 1-21. doi: 10.1007/978-4-431-53979-7_1. [7] Laura Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Z., 268 (2011), 371-407. doi: 10.1007/s00209-010-0677-6. [8] Thierry Champion and Luigi De Pascale, The Monge problem for strictly convex norms in $\mathbbR^d$, J. Eur. Math. Soc. (JEMS), 12 (2010), 1355-1369. doi: 10.4171/JEMS/234. [9] ______, The Monge problem in $\mathbbR^d$, Duke Math. J., 157 (2011), 551-572. doi: 10.1215/00127094-1272939. [10] ______, The Monge problem in $\mathbbR^d$: Variations on a theme, Journal of Mathematical Sciences, 181 (2012), 856-866. doi: 10.1007/s10958-012-0719-1. [11] Thierry Champion, Luigi De Pascale and Petri Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20. doi: 10.1137/07069938X. [12] Dario Cordero-Erausquin, Sur le transport de mesures périodiques, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 199-202. doi: 10.1016/S0764-4442(00)88593-6. [13] Luis A. Caffarelli, Mikhail Feldman and Robert J. McCann, Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), 1-26 (electronic). doi: 10.1090/S0894-0347-01-00376-9. [14] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999). doi: 10.1090/memo/0653. [15] Albert Fathi and Alessio Figalli, Optimal transportation on non-compact manifolds, Israel J. Math., 175 (2010), 1-59. doi: 10.1007/s11856-010-0001-5. [16] Alessio Figalli, Existence, uniqueness, and regularity of optimal transport maps, SIAM J. Math. Anal., 39 (2007), 126-137. doi: 10.1137/060665555. [17] _______, The Monge problem on non-compact manifolds, Rend. Semin. Mat. Univ. Padova, 117 (2007), 147-166. [18] Wilfrid Gangbo and Robert J. McCann, Optimal maps in Monge's mass transport problem, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1653-1658. [19] _______, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620. [20] Nassif Ghoussoub and Abbas Moameni, A self-dual polar factorization for vector fields, Comm. Pure Appl. Math., 66 (2013), 905-933. doi: 10.1002/cpa.21430. [21] L. V. Kantorovith, On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS (N. S.), 37 (1942), 199-201. [22] _______, On a problem of Monge, (in Russian) Uspekhi Mat. Nauk., 3 (1948), 225-226. [23] Eckehard Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Analysis, 9 (1985), 1381-1399. doi: 10.1016/0362-546X(85)90097-5. [24] Vladimir Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-Valued Anal., 7 (1999), 7-32. doi: 10.1023/A:1008753021652. [25] Robert J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679. [26] Gaspard Monge, "Mémoire sur la Théorie des Déblais et des Remblais," Histoire de l'Académie des Sciences de Paris, (1781). [27] A. Pratelli, On the sufficiency of $c$-cyclical monotonicity for optimality of transport plans, Math. Z., 258 (2008), 677-690. doi: 10.1007/s00209-007-0191-7. [28] L. Rüschendorf and S. T. Rachev, A characterization of random variables with minimum $L^2$-distance, J. Multivariate Anal., 32 (1990), 48-54. doi: 10.1016/0047-259X(90)90070-X. [29] Ludger Rüschendorf, On $c$-optimal random variables, Statist. Probab. Lett., 27 (1996), 267-270. doi: 10.1016/0167-7152(95)00078-X. [30] Walter Schachermayer and Josef Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proc. Amer. Math. Soc., 137 (2009), 519-529. doi: 10.1090/S0002-9939-08-09419-7. [31] Neil S. Trudinger and Xu-Jia Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations, 13 (2001), 19-31. doi: 10.1007/PL00009922. [32] Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1007/b12016. [33] _______, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
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