Random Conical Tessellations

We consider tessellations of the Euclidean (d− 1)-sphere by (d− 2)-dimensional great subspheres or, equivalently, tessellations of Euclidean d-space by hyperplanes through the origin; these we call conical tessellations. For random polyhedral cones defined as typical cones in a conical tessellation by random hyperplanes, and for their dual cones, we study expectations of certain geometric functionals and extend formulas of Cover and Efron (1967), by including additional geometric functionals, among them the conical intrinsic volumes. For isotropic conical tessellations, we find counterparts to results of Miles (1961); they can be interpreted as giving the covariance matrix for a random vector whose components are total k-face contents of certain spherical random polytopes.