Caffarelli's contraction theorem and related estimates on Brenier maps

This thesis focuses on global estimates on the Brenier map, the optimal transport map for the quadratic cost. We start with an overview of the classical theory of optimal transport, up to the characterization of Brenier maps as gradients of convex functions. We then introduce the Monge-Ampère equation, a powerful tool throughout the study of the regularity of optimal maps, and mention the main results in this direction. Finally, we focus on Caffarelli's contraction theorem, which states that the Brenier map from a Gaussian measure to a log-concave perturbation is 1-Lipschitz; this result allows several functional inequalities to be transferred from the Gaussian setting with no loss in the optimal constant. We provide the proofs of Caffarelli's theorem and its recently established Laplacian counterpart; next, we show that both results are examples of a single phenomenon, and prove that an analogous bound holds for any convex, increasing and positively homogeneous function of the Hessian. This part is an original contribution based on the author's preprint.