Boundary density and Voronoi set estimation for irregular sets

Lachièze-Rey, Raphaël; Vega, Sergio

doi:10.1090/tran/6848

Boundary density and Voronoi set estimation for irregular sets
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by Raphaël Lachièze-Rey and Sergio Vega PDF

Trans. Amer. Math. Soc. 369 (2017), 4953-4976 Request permission

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, 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-Esseen bounds in $n^{-s/2d}$ 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.

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