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Boundary density and Voronoi set estimation for irregular sets


Authors: Raphaël Lachièze-Rey and Sergio Vega
Journal: Trans. Amer. Math. Soc. 369 (2017), 4953-4976
MSC (2010): Primary 60D05, 60F05, 28A80; Secondary 28A78, 49Q15
DOI: https://doi.org/10.1090/tran/6848
Published electronically: February 13, 2017
MathSciNet review: 3632556
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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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Additional Information

Raphaël Lachièze-Rey
Affiliation: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, France
Email: raphael.lachieze-rey@parisdescartes.fr

Sergio Vega
Affiliation: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, France

DOI: https://doi.org/10.1090/tran/6848
Keywords: Voronoi approximation, set estimation, Minkowski dimension, Berry-Esseen bounds, self-similar sets
Received by editor(s): January 20, 2015
Received by editor(s) in revised form: January 21, 2015, and August 19, 2015
Published electronically: February 13, 2017
Article copyright: © Copyright 2017 American Mathematical Society