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.
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Transactions of the American Mathematical Society, American Mathematical Society, 2017, 369, pp.4953-4976. 〈http://www.ams.org/journals/tran/2017-369-07/S0002-9947-2017-06848-3/home.html〉
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https://hal.archives-ouvertes.fr/hal-01105205
Contributeur : Raphael Lachieze-Rey <>
Soumis le : lundi 23 novembre 2015 - 23:48:12
Dernière modification le : mardi 4 décembre 2018 - 13:14:05
Document(s) archivé(s) le : vendredi 28 avril 2017 - 16:25:29

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

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Raphaël Lachièze-Rey, Sergio Vega. Boundary density and Voronoi set estimation for irregular sets. Transactions of the American Mathematical Society, American Mathematical Society, 2017, 369, pp.4953-4976. 〈http://www.ams.org/journals/tran/2017-369-07/S0002-9947-2017-06848-3/home.html〉. 〈hal-01105205v3〉

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