Faces in random great hypersphere tessellations

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 Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as n → ∞ 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.

Random inscribed polytopes in projective geometries

“…This continues a recent trend in stochastic geometry of generalizing known results to the non-Euclidean setting, and in particular to spherical and hyperbolic geometry, see e.g. [5,6,24,28,29,32,33,34,35,36,37,38].…”