Duality for Borel measurable cost functions
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- by Mathias Beiglböck and Walter Schachermayer
- Trans. Amer. Math. Soc. 363 (2011), 4203-4224
- DOI: https://doi.org/10.1090/S0002-9947-2011-05174-3
- Published electronically: March 14, 2011
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Abstract:
We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\times Y\to [0,\infty )$ is an arbitrary Borel measurable cost function on the product of Polish spaces $X,Y$. In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.References
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Bibliographic Information
- Mathias Beiglböck
- Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
- Email: mathias.beiglboeck@univie.ac.at
- Walter Schachermayer
- Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
- Email: walter.schachermayer@univie.ac.at
- Received by editor(s): August 7, 2008
- Received by editor(s) in revised form: July 11, 2009
- Published electronically: March 14, 2011
- Additional Notes: The first author gratefully acknowledges financial support from the Austrian Science Fund (FWF) under grants S9612 and P21209. The second author gratefully acknowledges financial support from the Austrian Science Fund (FWF) under grant P19456, from the Vienna Science and Technology Fund (WWTF) under grant MA13 and from the Christian Doppler Research Association (CDG)
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 4203-4224
- MSC (2010): Primary 49K27, 28A05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05174-3
- MathSciNet review: 2792985