  Given a determinate (multivariate) probability measure $\mu$, we characterize
Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance
$W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on
the parameters $(m,\Sigma)$ of the Gaussians is supported on a compact set
$S$.(i) We first show that such mixtures are optimal solutions of a particular
optimal transport (OT) problem where the marginal $\nu\_{\phi}$ of the OT
problem is also unknown via the mixing measure variable $\phi$. Next (ii) by
using a well-known specific property of Gaussian measures, this optimal
transport is then viewed as a Generalized Moment Problem (GMP) and if the set
$S$ of mixture parameters $(m,\Sigma)$ is a basic compact semi-algebraic set,
we provide a "mesh-free" numerical scheme to approximate as closely as desired
the optimal distance by solving a hierarchy of semidefinite relaxations of
increasing size. In particular, we neither assume that the mixing measure is
finitely supported nor that the variance is the same for all components. If the
original measure $\mu$ is not a Gaussian mixture with parameters $(m,\Sigma)\in
S$, then a strictly positive distance is detected at a finite step of the
hierarchy. If the original measure $\mu$ is a Gaussian mixture with parameters
$(m,\Sigma)\in S$, then all semidefinite relaxations of the hierarchy have same
zero optimal value. Moreover if the mixing measure is atomic with finite
support, its components can sometimes be extracted from an optimal solution at
some semidefinite relaxation of the hierarchy when Curto & Fialkow's flatness
condition holds for some moment matrix.
