The optimal transport problem and its applications

Abstract

[en] The objective of this project is to present the base of Optimal Transport Theory and some of its applications. The Optimal Transport Problem was first studied by Monge in the 18th century, and later reformulated by Kantorovich during the 20th century, being this second version the main object of study. One of the key results relating Monge’s formulation is Brenier’s theorem, which we will prove and apply to prove the Isoperimetric inequality and the Sobolev inequality. By employing a different method we will prove another classical result, the Brunn-Minkowski inequality. This essay concludes with some conditions for the two problems to have the same optimal value. The other main topic studied during this work are the Wasserstein spaces. They are a family of probability measures spaces where we use Optimal transport to construct a metric, the Wasserstein distance. A key result is that it metrizes the weak topology of these spaces.