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