The question of approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the rate of convergence as the number of points tends to infinity, depending on the moment parameter, the parameter in the Wasserstein distance, and the dimension. In certain low-dimensional regimes and for measures with unbounded support, the rates are improvements over those obtained through other methods, including through random sampling. Except for some critical cases, the rates are shown to be optimal.
Benjamin Seeger (Fri,) studied this question.