  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.
