# 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 $s>d-1$ 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.
Keywords :
Type de document :
Pré-publication, Document de travail
MAP5 2015-03. to appear in Trans. AMS. 2015
Domaine :

https://hal.archives-ouvertes.fr/hal-01105205
Contributeur : Raphael Lachieze-Rey <>
Soumis le : lundi 23 novembre 2015 - 23:48:12
Dernière modification le : mardi 11 octobre 2016 - 12:00:22

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selfsimilar-2d-submission.pdf
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### Identifiants

• HAL Id : hal-01105205, version 3
• ARXIV : 1501.04724

### Citation

Raphaël Lachièze-Rey, Sergio Vega. Boundary density and Voronoi set estimation for irregular sets. MAP5 2015-03. to appear in Trans. AMS. 2015. <hal-01105205v3>

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