On Weak Convergence of Empirical Measures for Sets Minimizing the Hausdorff Distance

Abstract

This paper investigates the asymptotic behavior of finite sets that minimize the Hausdorff distance to a compact metric space. The primary objective is to establish conditions under which the empirical measures supported on these optimal sets converge weakly to the normalized Hausdorff measure. To facilitate this, we introduce the class of uniformly asymptotically open convex Euclidean metrics. These metrics characterize spaces that exhibit a local Euclidean structure in an asymptotic sense, allowing for a rigorous analysis of the geometry of Voronoi cells. The paper provides a sufficient reformulation of weak convergence based on the concept of sets uniformly eating Voronoi boundaries with respect to a regular measure. A complete proof of weak convergence to the normalized one-dimensional Hausdorff measure is presented for the specific case of a connected one-dimensional manifold (the circle). While the one-dimensional case yields a clear result due to the simple structure of Voronoi boundaries, the paper concludes by noting that higher-dimensional cases remain an open challenge due to the increased geometric complexity of the resulting Voronoi cells.