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Optimal transport and the geometry of $ L^{1}(\mathbb{R}^d)$


Authors: Ivar Ekeland and Walter Schachermayer
Journal: Proc. Amer. Math. Soc. 142 (2014), 3585-3596
MSC (2010): Primary 46B20, 46B25; Secondary 65K10
DOI: https://doi.org/10.1090/S0002-9939-2014-12094-6
Published electronically: July 2, 2014
MathSciNet review: 3238434
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Abstract: A classical theorem due to R. Phelps states that if $ C$ is a weakly compact set in a Banach space $ E$, the strongly exposing functionals form a dense subset of the dual space $ E^{\prime }$. In this paper, we look at the concrete situation where $ C\subset L^{1}(\mathbb{R}^{d})$ is the closed convex hull of the set of random variables $ Y\in L^{1}(\mathbb{R}^{d})$ having a given law $ \nu $. Using the theory of optimal transport, we show that every random variable $ X\in L^{\infty }(\mathbb{R}^{d})$, the law of which is absolutely continuous with respect to the Lebesgue measure, strongly exposes the set $ C$. Of course these random variables are dense in $ L^{\infty }(\mathbb{R}^{d})$.


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Additional Information

Ivar Ekeland
Affiliation: Department of Mathematics, University of British Columbia, 121–1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2

Walter Schachermayer
Affiliation: Faculty of Mathematics, University of Vienna, Room 06.131, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria
Email: walter.schachermayer@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9939-2014-12094-6
Received by editor(s): May 15, 2012
Received by editor(s) in revised form: November 11, 2012
Published electronically: July 2, 2014
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society

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