Optimal transport and geometric inequalities

Abstract

In this thesis we will first introduce some important concepts connected to optimal transport, including the Brenier map, a map from R n to R n derived from a convex potential pushing forward one probability measure to another, and the Monge-Ampère equation, a partial differential equation, linking the densities of these measures and the Brenier map. The second and main part presents proofs of several important geometric and analytic inequalities, namely the Brunn-Minkowski inequality, the Prèkopa-Leindler inequality, the Minkowski inequality, the (reverse) Brascamp-Lieb inequality and the Gagliardo-Nirenberg-Sobolev inequality, which are based on the aforementioned tools of optimal transportation.