#### Vol. 8, No. 1, 2015

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Eigenvalue distribution of optimal transportation

### Bo’az B. Klartag and Alexander V. Kolesnikov

Vol. 8 (2015), No. 1, 33–55
##### 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 $n$.

##### Keywords
transportation of measure, log-concave measures
Primary: 35J96
##### Milestones
Received: 13 February 2014
Accepted: 2 December 2014
Published: 15 April 2015
##### Authors
 Bo’az B. Klartag School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv Israel Alexander V. Kolesnikov Faculty of Mathematics National Research University Higher School of Economics 117312, Moscow Vavilova St., 7 Russia