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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterization of optimal transport plans for the Monge-Kantorovich problem


Authors: Walter Schachermayer and Josef Teichmann
Journal: Proc. Amer. Math. Soc. 137 (2009), 519-529
MSC (2000): Primary 49J45, 28A35
Published electronically: September 9, 2008
MathSciNet review: 2448572
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Abstract: We prove that $ c$-cyclically monotone transport plans $ \pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani's book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature.


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Walter Schachermayer
Affiliation: Technical University Vienna, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria

Josef Teichmann
Affiliation: Technical University Vienna, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09419-7
PII: S 0002-9939(08)09419-7
Received by editor(s): February 15, 2006
Received by editor(s) in revised form: August 24, 2007
Published electronically: September 9, 2008
Additional Notes: Financial support from the Austrian Science Fund (FWF) under grant P 15889, from the Vienna Science Foundation (WWTF) under grant MA13, and from the European Union under grant HPRN-CT-2002-00281 is gratefully acknowledged. Furthermore this work was financially supported by the Christian Doppler Research Association (CDG). The authors gratefully acknowledge a fruitful collaboration with and continued support by Bank Austria through CDG
Communicated by: David Preiss
Article copyright: © Copyright 2008 American Mathematical Society