1. Faces in random great hypersphere tessellations
- Author
-
Christoph Thäle and Zakhar Kabluchko
- Subjects
Statistics and Probability ,Unit sphere ,Dimension (graph theory) ,spherical Quermaßintegral ,$f$-vector ,Polytope ,statistical dimension ,52A55 ,52B11 ,typical spherical face ,Intersection ,Mathematics - Metric Geometry ,Euclidean geometry ,FOS: Mathematics ,Mathematics - Combinatorics ,spherical intrinsic volume ,60D05 ,Mathematics ,Mathematical analysis ,Isotropy ,Probability (math.PR) ,Metric Geometry (math.MG) ,Hypersphere ,52A22 ,great hypersphere tessellation ,Distribution (mathematics) ,spherical stochastic geometry ,Primary 52A22, 60D05, Secondary 52A55, 52B11 ,Combinatorics (math.CO) ,Statistics, Probability and Uncertainty ,intersection probability ,weighted spherical face ,Mathematics - Probability - Abstract
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 Querma\ss integrals 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.
- Published
- 2020
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