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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2017

Boundary density and Voronoi set estimation for irregular sets

Résumé

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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Dates et versions

hal-01105205 , version 1 (20-01-2015)
hal-01105205 , version 2 (21-01-2015)
hal-01105205 , version 3 (23-11-2015)

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Raphaël Lachièze-Rey, Sergio Vega. Boundary density and Voronoi set estimation for irregular sets. Transactions of the American Mathematical Society, 2017, 369, pp.4953-4976. ⟨hal-01105205v3⟩
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