Eigenvalue distribution of optimal transportation

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