Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function

We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the $L^p$-Wasserstein distance). We first propose a criterion on the quantization level $N$, valid for any norm on $\mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving the Vorono\"i diagram. Then, we prove that in the $L^2$-case on a (separable) Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is optimal. More quantization based characterization cases on dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found in the end of this paper.