Boundary density and Voronoi set estimation for irregular sets

Author:

Lachièze-Rey Raphaël,Vega Sergio

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.

Funder

Ministère de l'Enseignement Supérieur et de la Recherche

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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1. Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications;Journal of Scientific Computing;2022-07-05

2. Normal approximation for stabilizing functionals;The Annals of Applied Probability;2019-04-01

3. New Berry–Esseen bounds for functionals of binomial point processes;The Annals of Applied Probability;2017-08-01

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