Boundary density and Voronoi set estimation for irregular sets

Abstract

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension s > d − 1 s>d-1 in R d \mathbb {R}^{d} . These quantities turn out to be crucial in some problems of set estimation, as we show here for the Voronoi approximation of the set with a random input constituted by n n iid points in some larger bounded domain. We prove that some classes of such sets have positive inner and outer boundary density, and therefore satisfy Berry-Esseen bounds in n − s / 2 d n^{-s/2d} for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counterexample. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.