Eigenvalue distribution of optimal transportation

Abstract

We investigate the Brenier map $\nabla \Phi$ between the uniform measures on two convex domains in ${ℝ}^n$, or, more generally, between two log-concave probability measures on ${ℝ}^n$. We show that the eigenvalues of the Hessian matrix $D^2\Phi$ exhibit concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension $n$.