Boundary density and Voronoi set estimation for irregular sets

TL;DR

This paper investigates boundary densities of self-similar sets with Minkowski dimension greater than d-1, analyzing their impact on Voronoi set estimation and convergence rates in set approximation.

Contribution

It introduces boundary density concepts for complex sets and demonstrates their significance in Voronoi approximation, providing bounds and convergence rates.

Findings

Sets with self-similar boundaries have positive boundary densities.

Boundary densities influence the convergence rates of set estimations.

Examples include the Von Koch flake and a Cantor boundary set.

Abstract

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension $s\textgreater{}d-1$ in $\mathbb{R}^{d}$. These quantities turn out to be crucial in some problems of set estimation theory, as we show here for the Voronoi approximation of the set with a random input constituted by $n$ 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-Essen bounds in $n^{-s/2d}$ for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counter-example. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.