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A generalized dual maximizer for the Monge–Kantorovich transport 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 general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y →  [0,∞]  is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic 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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