### ?

## Eigenvalue distribution of optimal transportation

Analysis and PDE. 2015. Vol. 8. No. 1.

B.B. Klartag, Kolesnikov A.

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.