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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Duality for Borel measurable cost functions
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by Mathias Beiglböck and Walter Schachermayer PDF
Trans. Amer. Math. Soc. 363 (2011), 4203-4224 Request permission

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
  • 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