Résumé en anglais

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})$