Error estimates for deterministic empirical approximations of probability measures

The question of optimally approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the optimal approximation distance, with an explicit rate of convergence to $0$ as the number of points tends to infinity that depends on the moment order, 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.