### Article

## Eigenvalue distribution of optimal transportation

We investigate the Brenier map ΔΦ between the uniform measures on two convex domains in ℝn, or, more generally, between two log-concave probability measures on ℝn. We show that the eigenvalues of the Hessian matrix D2Φ exhibit concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n. © 2015 Mathematical Sciences Publishers.