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 log-concave 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 $n$.

23 pages

References in corpus (2)

• Mass transportation and contractions
• Logarithmically-concave moment measures I

Cited by in corpus (4)

• Optimal Concentration of Information Content For Log-Concave Densities
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• Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
• Extremal Kaehler-Einstein metric for two-dimensional convex bodies