A recent paper by Cordero-Erausquin and Klartag provides a characterization
of the measures μ on Rd 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: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−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