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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
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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
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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