Consider the Brenier map between the uniform measures on two convex domains, or, more generally, between two log-concave probability measures on R^n. It is well-known that the Brenier map is the gradient of a convex function F. In this lecture we will investigate the eigenvalues of the Hessian of F, and we will show that they exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n. Joint work with A. Kolesnikov.