Optimal transport and the geometry of $L^{1}(\mathbb {R}^d)$
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- by Ivar Ekeland and Walter Schachermayer PDF
- Proc. Amer. Math. Soc. 142 (2014), 3585-3596 Request permission
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 $L^{\infty }(\mathbb {R}^{d})$.References
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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
- MR Author ID: 62405
- 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
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3238434