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2021 Faces in random great hypersphere tessellations
Zakhar Kabluchko, Christoph Thäle
Electron. J. Probab. 26: 1-35 (2021). DOI: 10.1214/20-EJP570


The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to \infty $ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.


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Zakhar Kabluchko. Christoph Thäle. "Faces in random great hypersphere tessellations." Electron. J. Probab. 26 1 - 35, 2021.


Received: 2 August 2020; Accepted: 6 December 2020; Published: 2021
First available in Project Euclid: 7 January 2021

Digital Object Identifier: 10.1214/20-EJP570

Primary: 52A22 , 60D05
Secondary: 52A55 , 52B11

Keywords: $f$-vector , great hypersphere tessellation , intersection probability , spherical intrinsic volume , spherical Quermaßintegral , spherical stochastic geometry , statistical dimension , typical spherical face , weighted spherical face


Vol.26 • 2021
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