Boundary density and Voronoi set estimation for irregular sets
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- by Raphaël Lachièze-Rey and Sergio Vega PDF
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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.References
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Additional Information
- Raphaël Lachièze-Rey
- Affiliation: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, France
- MR Author ID: 937531
- Email: raphael.lachieze-rey@parisdescartes.fr
- Sergio Vega
- Affiliation: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité, France
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3632556