OPTIMAL TRANSPORT AND THE GEOMETRY OF L 1 (R d )

Walter Schachermayer
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Walter Schachermayer: Universität Wien = University of Vienna

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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 0. In this paper, we look at the concrete situation where C L 1 (R d) is the closed convex hull of the set of random variables Y 2 L 1 (R d) having a given law. Using the theory of optimal transport, we show that every random variable X 2 L 1 (R d), the law of which is absolutely continuous with respect to Lebesgue measure, strongly exposes the set C. Of course these random variables are dense in L 1 (R d).

Date: 2014
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Published in Proceedings of the American Mathematical Society, 2014, 142, pp.3585-3596. ⟨10.1090/S0002-9939-2014-12094-6⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01521488

DOI: 10.1090/S0002-9939-2014-12094-6

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