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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Face numbers of high-dimensional Poisson zero cells
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by Zakhar Kabluchko
Proc. Amer. Math. Soc. 151 (2023), 401-415
Published electronically: August 18, 2022


Let $\mathcal Z_d$ be the zero cell of a $d$-dimensional, isotropic and stationary Poisson hyperplane tessellation. We study the asymptotic behavior of the expected number of $k$-dimensional faces of $\mathcal Z_d$, as $d\to \infty$. For example, we show that the expected number of hyperfaces of $\mathcal Z_d$ is asymptotically equivalent to $\sqrt {2\pi /3}\, d^{3/2}$, as $d\to \infty$. We also prove that the expected solid angle of a random cone spanned by $d$ random vectors that are independent and uniformly distributed on the unit upper half-sphere in $\mathbb R^{d}$ is asymptotic to $\sqrt 3 \pi ^{-d}$, as $d\to \infty$.
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Bibliographic Information
  • Zakhar Kabluchko
  • Affiliation: Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149 Münster, Germany
  • MR Author ID: 696619
  • ORCID: 0000-0001-8483-3373
  • Email:
  • Received by editor(s): October 21, 2021
  • Received by editor(s) in revised form: March 17, 2022
  • Published electronically: August 18, 2022
  • Additional Notes: The work was supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure and by the DFG priority program SPP 2265 Random Geometric Systems
  • Communicated by: Qi-Man Shao
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 401-415
  • MSC (2020): Primary 60D05, 52A23; Secondary 52A22, 52B11, 52B05, 60G55, 33E20
  • DOI:
  • MathSciNet review: 4504634