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

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 $L^{\infty }(\mathbb {R}^{d})$.