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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Characterization of optimal transport plans for the Monge-Kantorovich problem
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by Walter Schachermayer and Josef Teichmann
Proc. Amer. Math. Soc. 137 (2009), 519-529
DOI: https://doi.org/10.1090/S0002-9939-08-09419-7
Published electronically: September 9, 2008

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.
References
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Bibliographic Information
  • 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
  • MR Author ID: 654648
  • 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
  • © Copyright 2008 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 519-529
  • MSC (2000): Primary 49J45, 28A35
  • DOI: https://doi.org/10.1090/S0002-9939-08-09419-7
  • MathSciNet review: 2448572