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Set Reconstruction by Voronoi Cells

Part of: Geometric probability and stochastic geometry Stochastic processes Combinatorial probability General convexity

Published online by Cambridge University Press:  04 January 2016

M. Reitzner ,
E. Spodarev  and
D. Zaporozhets
M. Reitzner*
Affiliation:
Universität Osnabrück
E. Spodarev*
Affiliation:
Universität Ulm
D. Zaporozhets*
Affiliation:
V. A. Steklov Mathematical Institute
*
∗ Postal address: Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany. Email address: matthias.reitzner@uni-osnabrueck.de
∗∗ Postal address: Institut für Stochastik, Universität Ulm, 89069 Ulm, Germany. Email address: evgeny.spodarev@uni-ulm.de
∗∗∗ Postal address: St. Petersburg Department, V. A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia. Email address: zap1979@gmail.com
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Abstract

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For a Borel set A and a homogeneous Poisson point process η in of intensity λ>0, define the Poisson–Voronoi approximation A η of A as a union of all Voronoi cells with nuclei from η lying in A. If A has a finite volume and perimeter, we find an exact asymptotic of E Vol(AΔ A η) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(A η) and Vol(AΔ A η) together with their asymptotics for large λ are obtained as well.

Keywords

Poisson point processPoisson–Voronoi cellPoisson–Voronoi tessellationperimeter

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry 60C05: Combinatorial probability

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 938 - 953
Copyright
© Applied Probability Trust 

Footnotes

Partially supported by RFBR (10-01-00242), NSh-4472.2010.1, RFBR-DFG (09-0191331), and DFG (436 RUS 113/962/0-1 R) grants.

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