Research Article

A generalized dual maximizer for the Monge–Kantorovich transport problem ^∗

Mathias Beiglböck^a1, Christian Léonard^a2 and Walter Schachermayer^a3

^a1 University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria. mathias.beiglboeck@univie.ac.at

^a2 Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92001 Nanterre, France; christian.leonard@u-paris10.fr

^a3 University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria; walter.schachermayer@univie.ac.at

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.

(Received September 7 2010)

(Revised May 15 2011)

(Online publication July 16 2012)

Key Words:

• Optimal transport;
• duality in function spaces;
• Fenchel’s perturbation technique

Mathematics Subject Classification:

• 46E30;
• 46N10;
• 49J45;
• 28A35

Footnotes

^∗  The first author acknowledges financial support from the Austrian Science Fund (FWF) under grant P21209. The third author acknowledges support from the Austrian Science Fund (FWF) under grant P19456, from the Vienna Science and Technology Fund (WWTF) under grant MA13 and by the Christian Doppler Research Association (CDG). All authors thank A. Pratelli for helpful discussions on the topic of this paper. We also thank M. Goldstern and G. Maresch for their advice.