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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Duality for Borel measurable cost functions


Authors: Mathias Beiglböck and Walter Schachermayer
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
Published electronically: March 14, 2011
MathSciNet review: 2792985
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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.


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Additional 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

DOI: https://doi.org/10.1090/S0002-9947-2011-05174-3
Keywords: Monge-Kantorovich problem, Monge-Kantorovich duality, $c$-cyclical monotonicity, measurable cost function
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)
Article copyright: © Copyright 2011 American Mathematical Society