Calculus of Variations and Geometric Measure Theory

F. Santambrogio

Dealing with moment measures via entropy and optimal transport

created by santambro on 25 Jun 2015
modified on 19 Jun 2016


Accepted Paper

Inserted: 25 jun 2015
Last Updated: 19 jun 2016

Journal: J. Functional Analysis
Year: 2016


A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures $\mu$ on $\R^d$ which can be expressed as the moment measures of suitable convex functions $u$, i.e. are of the form $(\nabla u)_\#e^{- u}$ for $u:\R^d\to\R\cup\{+\infty\}$ and finds the corresponding $u$ by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities $\rho$ and the optimizer $\rho$ turns out to be $e^{-u}$. This requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport. The notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers.

Keywords: displacement convexity, entropy, log-concave measures, convexity constraints, Semi-continuity, toland duality