Advances in Applied Probability

Set reconstruction by Voronoi cells

M. Reitzner, E. Spodarev, and D. Zaporozhets

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For a Borel set A and a homogeneous Poisson point process η in ∝d of intensity λ>0, define the Poisson--Voronoi approximation Aη 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 Aη) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(Aη) and Vol(AΔ Aη) together with their asymptotics for large λ are obtained as well.

Article information

Adv. in Appl. Probab., Volume 44, Number 4 (2012), 938-953.

First available in Project Euclid: 5 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60C05: Combinatorial probability

Poisson point process Poisson--Voronoi cell Poisson--Voronoi tessellation perimeter


Reitzner, M.; Spodarev, E.; Zaporozhets, D. Set reconstruction by Voronoi cells. Adv. in Appl. Probab. 44 (2012), no. 4, 938--953. doi:10.1239/aap/1354716584.

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  • Ambrosio, L., Colesanti, A. and Villa, E. (2008). Outer Minkowski content for some classes of closed sets. Math. Ann. 342, 727–748.
  • Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York.
  • Averkov, G. and Bianchi, G. (2009). Confirmation of Matheron's conjecture on the covariogram of a planar convex body. J. Europ. Math. Soc. 11, 1187–1202.
  • Billingsley, P. (1979). Probability and Measure. John Wiley, New York.
  • Einmahl, J. H. J. and Khmaladze, E. V. (2001). The two-sample problem in $\mathbb{R}^m$ and measure-valued martingales. In State of the Art in Probability and Statistics (Leiden, 1999; IMS Lecture Notes Monogr. Ser. 36), Institute of Mathematical Statistics, Beachwood, OH, pp. 434–463.
  • Galerne, B. (2011). Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30, 39–51.
  • Heveling, M. and Reitzner, M. (2009). Poisson–Voronoi approximation. Ann. Appl. Prob. 19, 719–736.
  • Khmaladze, E. and Toronjadze, N. (2001). On the almost sure coverage property of Voronoi tessellation: the ${\mathbb{R}}^ 1$ case. Adv. Appl. Prob. 33, 756–764.
  • Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. Prob. Theory Relat. Fields 150, 663–690.
  • Møller, J. (1994). Lectures on Random Voronoĭ Tessellations (Lecture Notes Statist. 87). Springer, New York.
  • Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.
  • Schneider, R., and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Schulte, M. (2012). A central limit theorem for the Poisson–Voronoi approximation. Adv. Appl. Math. 49, 285–306.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.
  • Wu, L. (2000). A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Prob. Theory Relat. Fields 118, 427–438.