Face numbers of high-dimensional Poisson zero cells
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- by Zakhar Kabluchko
- Proc. Amer. Math. Soc. 151 (2023), 401-415
- DOI: https://doi.org/10.1090/proc/16085
- Published electronically: August 18, 2022
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Abstract:
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 $d\to \infty$.References
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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: zakhar.kabluchko@uni-muenster.de
- 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: https://doi.org/10.1090/proc/16085
- MathSciNet review: 4504634