Geodesic
Two ways to define compatible metrics on the simplex of measures
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 38-48 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We introduce two general methods of lifting a metric on a space to the simplex of probability measures on the metric space. The first one is the method of transportation plans, or the coupling method; the second one is the method of considering norms dual to the restrictions of the Lipschitz norm to subspaces. The intersection of these two classes of metrics consists of the Kantorovich metric. 
@article{ZNSL_2013_411_a1,
     author = {A. M. Vershik},
     title = {Two ways to define compatible metrics on the simplex of measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {38--48},
     year = {2013},
     volume = {411},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/}
}
TY  - JOUR
AU  - A. M. Vershik
TI  - Two ways to define compatible metrics on the simplex of measures
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 38
EP  - 48
VL  - 411
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/
LA  - ru
ID  - ZNSL_2013_411_a1
ER  - 
%0 Journal Article
%A A. M. Vershik
%T Two ways to define compatible metrics on the simplex of measures
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 38-48
%V 411
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/
%G ru
%F ZNSL_2013_411_a1
A. M. Vershik. Two ways to define compatible metrics on the simplex of measures. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 38-48. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/

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

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

[3] A. M. Vershik, P. B. Zatitskii, F. V. Petrov, “Virtual Continuity of Measurable Functions of Several Variables and Embedding Theorems”, Funct. Anal. Appl., 47:3 (2013), 165–173 | DOI | DOI | MR | Zbl

[4] L. V. Kantorovich, Matematicheskie metody organizatsii i planirovaniya proizvodstva, Izd-vo LGU, L., 1939

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

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

[7] L. V. Kantorovich, G. Sh. Rubinshtein, “Ob odnom funktsionalnom prostranstve i nekotorykh ekstremalnykh zadachakh”, Dokl. AN SSSR, 115:6 (1958), 52–59

[8] R. Holmes, “The universal separable metric space of Urysohn and isometric embeddings thereof in Banach spaces”, Fund. Math., 140:3 (1992), 199–223 | MR | Zbl

[9] J. Melleray, F. Petrov, A. Vershik, “Linearly rigid metric spaces and the embedding problem”, Fund. Math., 199:2 (2008), 177–194 | DOI | MR | Zbl

[10] F. Otto, “The geometry of dissipative evolution equations: the porous medium equation”, Comm. Partial Diff. Eqs., 26:1–2 (2001), 101–174 | DOI | MR | Zbl

[11] M. Rieffel, H. Li, “Metric aspects of noncommutative homogeneous spaces”, J. Funct. Anal., 257:7 (2009), 2325–2350 | DOI | MR

[12] V. Strassen, “The existence of probability measures with given marginals”, Ann. Math. Stat., 36:2 (1965), 423–439 | DOI | MR | Zbl

[13] A. Vershik, “Polymorphisms, Markov processes, and quasi-similarity”, Discrete Contin. Dyn. Syst., 13:5 (2005), 1305–1324 | DOI | MR | Zbl

[14] A. Vershik, “Long history of the Monge–Kantorovich transportation problem”, Math. Intelligencer, 35:4 (2013) (to appear) | DOI | MR

[15] C. Villani, Optimal Transport, Old and New, Springer, 2006 | MR