Comptes Rendus
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

Stefano Bianchini1 ; Laura Caravenna2
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, Volume 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
Stefano Bianchini 1 ; Laura Caravenna 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 
@article{CRMATH_2010__348_11-12_613_0,
     author = {Stefano Bianchini and Laura Caravenna},
     title = {On optimality of \protect\emph{c}-cyclically monotone transference plans},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {613--618},
     publisher = {Elsevier},
     volume = {348},
     number = {11-12},
     year = {2010},
     doi = {10.1016/j.crma.2010.03.022},
     language = {en},
}
TY  - JOUR
AU  - Stefano Bianchini
AU  - Laura Caravenna
TI  - On optimality of c-cyclically monotone transference plans
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 613
EP  - 618
VL  - 348
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2010.03.022
LA  - en
ID  - CRMATH_2010__348_11-12_613_0
ER  - 
%0 Journal Article
%A Stefano Bianchini
%A Laura Caravenna
%T On optimality of c-cyclically monotone transference plans
%J Comptes Rendus. Mathématique
%D 2010
%P 613-618
%V 348
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2010.03.022
%G en
%F CRMATH_2010__348_11-12_613_0
Stefano Bianchini; Laura Caravenna. On optimality of c-cyclically monotone transference plans. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 613-618. doi : 10.1016/j.crma.2010.03.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.022/

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

[2] L. Ambrosio; A. Pratelli 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] M. Beiglböck; M. Goldstern; G. Maresch; W. Schachermayer 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] S. Bianchini; L. Caravenna 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] D.H. Fremlin Measure Theory, vol. 1–4, Torres Fremlin, Colchester, 2001

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Nouvelle approche du théorème de ABB

Abdelhamid Bourass; Lahoussine Lafhim

C. R. Math (2009)