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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