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 Querma integrals 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.