no. 11-12
Mathematical Analysis/Calculus of Variations
On optimality of c-cyclically monotone transference plans
[Sur l'optimalité des plans de transport c-cycliques monotones]

Présenté par : Michel Talagrand

Bianchini, Stefano1 ; Caravenna, Laura2
1 SISSA, via Beirut 2, 34014 Trieste, Italy
2 CRM De Giorgi, Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56100 Pisa, Italy
Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 613-618.

Dans la présente note nous décrivons brièvement la construction introduite dans Bianchini and Caravenna (2009) [7] à propos de l'équivalence entre l'optimalité d'un plan de transport pour le problème de Monge–Kantorovich et la condition de monotonie c-cyclique—ainsi que d'autres sujets que cela nous amène à aborder. Nous souhaitons mettre en évidence l'hypothèse de mesurabilité sur la structure sous-jacente de pré-ordre linéaire.

This Note deals with the equivalence between the optimality of a transport plan for the Monge–Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in Bianchini and Caravenna (2009) [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems among the family of transport plans.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.03.022
Bianchini, Stefano 1 ; Caravenna, Laura 2

1 SISSA, via Beirut 2, 34014 Trieste, Italy
2 CRM De Giorgi, Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56100 Pisa, Italy 
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Bianchini, Stefano; Caravenna, Laura. On optimality of c-cyclically monotone transference plans. Comptes Rendus. Mathématique, Tome 348 (2010) no. 11-12, pp. 613-618. doi : 10.1016/j.crma.2010.03.022. http://www.numdam.org/articles/10.1016/j.crma.2010.03.022/

[1] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000

[2] Ambrosio, L.; Pratelli, A. Existence and Stability Results in the L1 Theory of Optimal Transportation, Optimal Transportation and Applications, Lecture Notes in Mathematics, vol. 1813, Springer, Berlin, 2001

[3] Beiglböck, M.; Goldstern, M.; Maresch, G.; Schachermayer, W. Optimal and better transport plans, J. Funct. Anal., Volume 256 (2009) no. 6, pp. 1907-1927

[4] M. Beiglböck, C. Leonard, W. Schachermayer, A general duality theorem for the Monge–Kantorovich transport problem

[5] M. Beiglböck, C. Leonard, W. Schachermayer, On the duality theory for the Monge–Kantorovich transport problem

[6] M. Beiglböck, W. Schachermayer, Duality for Borel measurable cost function, preprint

[7] Bianchini, S.; Caravenna, L. On the extremality, uniqueness and optimality of transference plans, Bull. Inst. Math. Acad. Sin. (N.S.), Volume 4 (2009) no. 4, pp. 353-455

[8] Fremlin, D.H. Measure Theory, vol. 1–4, Torres Fremlin, Colchester, 2001

[9] Harrington, L.; Marker, D.; Shelah, S. Borel orderings, Trans. Amer. Math. Soc., Volume 310 (1988) no. 1, pp. 293-302

[10] Hestir, K.; Williams, S.C. Supports of doubly stochastic measures, Bernoulli, Volume 1 (1995) no. 3, pp. 217-243

[11] Kanovei, V. When a partial Borel order is linearizable, Fund. Math., Volume 155 (1998) no. 3, pp. 301-309

[12] Kellerer, H.G. Duality theorems for marginals problems, Z. Wahrsch. Verw. Gebiete, Volume 67 (1984) no. 4, pp. 399-432

[13] Pratelli, A. On the sufficiency of c-cyclical monotonicity for optimality of transport plans, Math. Z. (2007)

[14] W. Schachermayer, J. Teichmann, Solution of a problem in Villani's book, preprint, 2005

[15] Schachermayer, W.; Teichmann, J. Characterization of optimal transport plans for the Monge–Kantorovich problem, Proc. Amer. Math. Soc., Volume 137 (2009) no. 2, pp. 519-529

[16] Smith, C.; Knott, M. On Hoeffding–Fréchet bounds and cyclic monotone relations, J. Multivariate Anal., Volume 40 (1992) no. 2, pp. 328-334

[17] Villani, C. Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, AMS, Providence, RI, 2003

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