Vol. 8, No. 1, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Φ between the uniform measures on two convex domains in n, or, more generally, between two log-concave probability measures on n. 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.

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