Eigenvalue distribution of optimal transportation

Klartag, Bo’az; Kolesnikov, Alexander

doi:10.2140/apde.2015.8.33

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Eigenvalue distribution of optimal transportation

Bo’az B. Klartag and Alexander V. Kolesnikov

Vol. 8 (2015), No. 1, 33–55

DOI: 10.2140/apde.2015.8.33

Abstract

We investigate the Brenier map $\nabla Φ$ 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 $D^{2} Φ$ 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

Mathematical Subject Classification 2010

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

Vol. 8, No. 1, 2015