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A generalized dual maximizer for the Monge–Kantorovichtransport problem∗

Published online by Cambridge University Press:  16 July 2012

Mathias Beiglböck ,
Christian Léonard  and
Walter Schachermayer
Mathias Beiglböck
Affiliation:
University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria. mathias.beiglboeck@univie.ac.at
Christian Léonard
Affiliation:
Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92001 Nanterre, France; christian.leonard@u-paris10.fr
Walter Schachermayer
Affiliation:
University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria; walter.schachermayer@univie.ac.at
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Abstract

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a generalsetting. The spaces X,Y are assumed to be polish and equipped with Borelprobability measures μ and ν. The transport costfunction c : X × Y →  [0,∞]  is assumedto be Borel measurable. We show that a dual optimizer always exists, provided we interpretit as a projective limit of certain finitely additive measures. Our methods are functionalanalytic and rely on Fenchel’s perturbation technique.

Keywords

Optimal transportduality in function spacesFenchel’s perturbation technique
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 306 - 323
Copyright
© EDP Sciences, SMAI, 2012

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