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Working paper

Eigenvalue distribution of optimal transportation.

arxiv.org. math. Cornell University, 2013. No. 1402.2636.
Kolesnikov A., Klartag B.
We investigate the Brenier map \nabla \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 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n.