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A recent paper by Cordero-Erausquin and Klartag provides a characterizationof the measures $\mu$ on $\R^d$ which can be expressed as the moment measuresof 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 avariational method in the class of convex functions. Here we propose a purelyoptimal-transport-based method to retrieve the same result. The variationalproblem becomes the minimization of an entropy and a transport cost amongdensities $\rho$ and the optimizer $\rho$ turns out to be $e^{-u}$. Thisrequires to develop some estimates and some semicontinuity results for thecorresponding functionals which are natural in optimal transport. The notion ofdisplacement convexity plays a crucial role in the characterization anduniqueness of the minimizers.

Dealing with moment measures via entropy and optimal transport