## Characterization of optimal transport plans for the Monge-Kantorovich problem

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- by Walter Schachermayer and Josef Teichmann PDF
- Proc. Amer. Math. Soc.
**137**(2009), 519-529 Request permission

## 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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## Additional 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