@unpublished{20570,
  abstract     = {We investigate the minimal error in approximating a general probability
measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with
prescribed cardinality $n$. The error is measured in the $p$-Wasserstein
distance. In particular, when $1\le p<d$, we establish asymptotic upper and
lower bounds as $n \to \infty$ on the rescaled minimal error that have the
same, explicit dependency on $\mu$.
  In some instances, we prove that the rescaled minimal error has a limit.
These include general measures in dimension $d = 2$ with $1 \le p < 2$, and
uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform
measures, we prove the limit existence for $p \ge d$ as well.
  For a class of compactly supported measures with H\"older densities, we
determine the convergence speed of the minimal error for every $p \ge 1$.
  Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper
estimate of the minimal error when $1 \le p < d$.
  In the initial sections, we survey the state of the art and draw connections
with similar problems, such as classical and random quantization.},
  author       = {Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {optimal empirical quantization, vector quantization, Wasserstein distance, semidiscrete optimal transport, Zador’s Theorem, Pierce’s Lemma},
  title        = {{Asymptotics for optimal empirical quantization of measures}},
  doi          = {10.48550/arXiv.2408.12924},
  year         = {2024},
}