Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-28T08:46:36.076Z Has data issue: false hasContentIssue false
  • English
  • Français

The $\beta$-Delaunay tessellation: Description of the model and geometry of typical cells

Part of: Geometric probability and stochastic geometry Stochastic processes Global differential geometry Polytopes and polyhedra General convexity

Published online by Cambridge University Press:  01 August 2022

Anna Gusakova ,
Zakhar Kabluchko  and
Christoph Thäle
Anna Gusakova*
Affiliation:
Münster University
Zakhar Kabluchko*
Affiliation:
Münster University
Christoph Thäle*
Affiliation:
Ruhr University Bochum
*
*Postal address: University of Münster, Mathematics Münster, Einsteinstrasse 62, 48149 Münster, Germany.
*Postal address: University of Münster, Mathematics Münster, Einsteinstrasse 62, 48149 Münster, Germany.
****Postal address: Ruhr University Bochum, Universitäatsstrasse 150, 44801 Bochum, Germany. Email address: christoph.thaele@rub.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce two new classes of stationary random simplicial tessellations, the so-called $\beta$- and $\beta^{\prime}$-Delaunay tessellations. Their construction is based on a space–time paraboloid hull process and generalizes that of the classical Poisson–Delaunay tessellation. We explicitly identify the distribution of volume-power-weighted typical cells, establishing thereby a remarkable connection to the classes of $\beta$- and $\beta^{\prime}$-polytopes. These representations are used to determine the principal characteristics of such cells, including volume moments, expected angle sums, and cell intensities.

Keywords

Angle sumsβ-Delaunay tessellationβ-polytopeβ′-polytopeLaguerre tessellationparaboloid convexityparaboloid hull processPoisson point processPoisson–Delaunay tessellationPoisson–Voronoi tessellationrandom polytopestochastic geometryweighted typical cellzero cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 52A22: Random convex sets and integral geometry 52B11: $n$-dimensional polytopes 53C65: Integral geometry

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1252 - 1290
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aurenhammer, F., Klein, R. and Lee, D.-T. (2013). Voronoi Diagrams and Delaunay Triangulations. World Scientific, London.CrossRefGoogle Scholar
Baumstark, V. and Last, G. (2007). Some distributional results for Poisson–Voronoi tessellations. Adv. Appl. Prob. 39, 16–40.CrossRefGoogle Scholar
Blaszczyszyn, B., Haenggi, M., Keeler, P. and Mukherjee, S. (2018). Stochastic Geometry Analysis of Cellular Networks. Cambridge University Press.CrossRefGoogle Scholar
Boissonnat, J.-D. and Yvinec, M. (1998). Algorithmic Geometry. Cambridge University Press. Translated from the 1995 French original by Hervé Brönnimann.CrossRefGoogle Scholar
Calka, P., Schreiber, T. and Yukich, J. E. (2013). Brownian limits, local limits and variance asymptotics for convex hulls in the ball. Ann. Prob. 41, 50–108.CrossRefGoogle Scholar
Calka, P. and Yukich, J. E. (2015). Variance asymptotics and scaling limits for Gaussian polytopes. Prob. Theory Relat. Fields 163, 259–301.CrossRefGoogle Scholar
Cheng, S.-W., Dey, T. and Shewchuk, J. R. (2013). Delaunay Mesh Generation: Algorithms and Mathematical Analysis. CRC Press, Boca Raton.Google Scholar
Edelsbrunner, H. (2014). A Short Course in Computational Geometry and Topology. Springer, Cham.CrossRefGoogle Scholar
Götze, F., Kabluchko, Z. and Zaporozhets, D. (2021). Grassmann angles and absorption probabilities of Gaussian convex hulls. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 501, 126–148.Google Scholar
Grote, J., Kabluchko, Z. and Thäle, C. (2019). Limit theorems for random simplices in high dimensions. ALEA 16, 141–177.CrossRefGoogle Scholar
Gusakova, A. and Thäle, C. (2021). The volume of simplices in high-dimensional Poisson–Delaunay tessellations. Ann. H. Lebesgue 4, 121–153.CrossRefGoogle Scholar
Haenggi, M. (2012). Stochastic Geometry for Wireless Networks. Cambridge University Press.CrossRefGoogle Scholar
Hug, D. and Schneider, R. (2004). Large cells in Poisson–Delaunay tessellations. Discrete Comput. Geom. 31, 503–514.CrossRefGoogle Scholar
Kabluchko, Z. (2021). Angles of random simplices and face numbers of random polytopes. Adv. Math. 380, Paper No. 107612.CrossRefGoogle Scholar
Kabluchko, Z., Temesvari, D. and Thäle, C. (2019). Expected intrinsic volumes and facet numbers of random beta-polytopes. Math. Nachr. 292, 79–105.CrossRefGoogle Scholar
Kabluchko, Z. and Thäle, C. (2021). The typical cell of a Voronoi tessellation on the sphere. Discrete Comput. Geom. 66, 1330–1350.CrossRefGoogle Scholar
Kabluchko, Z., Thäle, C. and Zaporozhets, D. (2020). Beta polytopes and Poisson polyhedra: f-vectors and angles. Adv. Math. 374, Paper No. 107333.CrossRefGoogle Scholar
Kabluchko, Z. and Zaporozhets, D. (2018). Angles of the Gaussian simplex. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 476, 79–91.Google Scholar
Kallenberg, O. (2017). Random Measures, Theory and Applications. Springer, Cham.CrossRefGoogle Scholar
Last, G. and Penrose, M. (2018). Lectures on the Poisson Process. Cambridge University Press.Google Scholar
Lautensack, C. (2007). Random Laguerre tessellations. Doctoral Thesis, Universität Karlsruhe.Google Scholar
Lautensack, C. and Zuyev, S. (2008). Random Laguerre tessellations. Adv. Appl. Prob. 40, 630–650.CrossRefGoogle Scholar
Lo, D. S. H. (2017). Finite Element Mesh Generation. Taylor and Francis, Boca Raton.Google Scholar
Mecke, J. and Muche, L. (1995). The Poisson Voronoi tessellation. I. A basic identity. Math. Nachr. 176, 199–208.CrossRefGoogle Scholar
MØller, J. (1989). Random tessellations in $\mathbb{R}^{d}$ . Adv. Appl. Prob. 21, 37–73.CrossRefGoogle Scholar
MØller, J. (1994). Lectures on Random Voronoi Tessellations. Springer, New York.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.CrossRefGoogle Scholar
Preparata, F. P. and Shamos, M. (1985). Computational Geometry. Springer, New York.CrossRefGoogle Scholar
Schlottmann, M. (1993). Periodic and quasi-periodic Laguerre tilings. Internat. J. Modern Phys. B 7, 1351–1363.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. B. G. Teubner, Stuttgart.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Schreiber, T. and Yukich, J. E. (2008). Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points. Ann. Prob. 36, 363–396.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
The CGAL Project (2020). CGAL User and Reference Manual, 5.0.2 edn. CGAL Editorial Board.Google Scholar