Dealing with moment measures via entropy and optimal transport

A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ   on RdRd which can be expressed as the moment measures of suitable convex functions u  , i.e. are of the form (∇u)#e−u(∇u)#e−u for u:Rd→R∪{+∞}u:Rd→R∪{+∞} 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 ρ and the optimizer ρ   turns out to be e−ue−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.