Dealing with moment measures via entropy and optimal transport

TL;DR

This paper introduces an optimal transport approach to characterize measures as moment measures of convex functions, providing a new variational method based on entropy minimization and transport cost.

Contribution

It offers a purely optimal-transport-based method to characterize measures as moment measures, replacing previous variational techniques with entropy and transport cost minimization.

Findings

Optimal transport method successfully characterizes measures as moment measures.

Displacement convexity ensures uniqueness of minimizers.

Develops estimates and semicontinuity results for related functionals.

Abstract

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.