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A new approach to weak convergence of random cones and polytopes

Part of: General convexity Polytopes and polyhedra Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  11 August 2020

Zakhar Kabluchko ,
Daniel Temesvari  and
Christoph Thäle
Zakhar Kabluchko
Affiliation:
Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Münster, Germany e-mail: zakhar.kabluchko@uni-muenster.de
Daniel Temesvari
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wien, Austria e-mail: daniel.temesvari@tuwien.ac.at
Christoph Thäle*
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
*
e-mail: christoph.thaele@rub.de
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Abstract

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated by n independent and uniformly distributed random linear hyperplanes in $\mathbb {R}^{d+1}$ , weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb {R}^d$ , as $n\to \infty $ .

Keywords

Cover–Efron conerandom conerandom polytoperandom tessellationScheffé’s lemmaSchläfli conestochastic geometrytypical cellweak convergence

MSC classification

Primary: 52A22: Random convex sets and integral geometry
Secondary: 60D05: Geometric probability and stochastic geometry 52A55: Spherical and hyperbolic convexity 52B11: $n$-dimensional polytopes 60F05: Central limit and other weak theorems
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 6 , December 2021 , pp. 1627 - 1647
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Z.K. has been supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure.

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