*Published Paper*

**Inserted:** 3 dec 2009

**Last Updated:** 1 jun 2011

**Journal:** JMPA

**Year:** 2011

**Abstract:**

Given a density function $f$ on an compact subset
of $\R^d$, we look at the problem of finding the best approximation of $f$ by discrete measures $\nu=\sum c_i \delta_{x_i}$ in the sense of the $p$-Wasserstein distance, subject to size constraints of the form $\sum h(c_i)\le \alpha$ where $h$ is a given weight
function (*entropy*). This is an important problem with applications
in economic planning of locations, in information theory and in shape optimization problems. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as $\alpha$ tends to infinity. In this paper, we introduce the relevant rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the $c_i$'s.
The asymptotic problem for large $\alpha$ can be then treated in the case of quite general entropy functions $h$.

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