Eigenvalue distribution of optimal transportation
Abstract
We investigate the Brenier map $\nabla \Phi$ between the uniform measures on two convex domains in $\mathbb{R}^n$ or more generally, between two logconcave 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 $n$.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.2636
 Bibcode:
 2014arXiv1402.2636K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 Mathematics  Metric Geometry
 EPrint:
 23 pages