Optimal mass transport and functional inequalities

Abstract

We formulate the optimal transportation problem, first with Monge's original question and then with Kantorovich's approach. We state Brenier's theorem and qe define fully-nonlinear Monge-Ampère type of partial differential equations. We use these tools together with the Arithmetic Mean-Geometric Mean inequality and Hölder's inequality in order to prove some important and well-known functional inequalities: the isoperimetric inequality and Sobolev inequalities such as Gagliardo-Nirenberg-Sobolev inequality. We deduce alternative statements for the isoperimetric inequality. We establish the GNS inequality for an arbitrary norm of R^n since the Euclidean structure plays no role on this approach. We also prove the GNS inequality and the isoperimetric inequality using classical tools without optimal transport techniques. Finally, we see that the Sobolev inequality and the isoperimetric inequality are equivalent in a compact n−dimensional Riemannian manifold.