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

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  • Walter Schachermayer

    (Universität Wien = University of Vienna)

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).

Suggested Citation

  • Walter Schachermayer, 2014. "OPTIMAL TRANSPORT AND THE GEOMETRY OF L 1 (R d )," Post-Print hal-01521488, HAL.
    • Handle: RePEc:hal:journl:hal-01521488
    DOI: 10.1090/S0002-9939-2014-12094-6
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