Intersection probabilities and kinematic formulas for polyhedral cones

Abstract

For polyhedral convex cones in R d , we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.

A new approach to weak convergence of random cones and polytopes

“…In other words, a polyhedral cone is just a set of solutions to a finite system of linear homogeneous inequalities. The study of random polyhedral cones has been initiated by Cover and Efron [11] and continued by Hug and Schneider [15] and Schneider [31]. Random spherical tessellations, i.e.…”

“…The Schläfli random cone S n is a polyhedral cone selected uniformly at random from this collection of cones ; see [15,31]. By definition, each cone has the same probability of 1/C(n, d + 1) to be selected.…”

“…, H n ) the collection of these cones. Following [15,31], the Schläfli random cone S n is a random closed cone picked uniformly at random from Cns(H 1 , . .…”

Faces in random great hypersphere tessellations

“…The natural question now arises whether there are mathematically tractable models for cones having a random shape that allow an exact determination of intersection probabilities. In this context, the following question has been studied in [30], which we rephrase in our equivalent spherical set-up. Our goal is to complement the result in [30] by studying the corresponding intersection probability for weighted typical cells.…”

“…Recent works directly linked with this text are the articles [6,18,23] on spherical convex hulls on half-spheres, the papers [9,10,11,14,16,17] dealing with different types of hyperplane or splitting tessellations in spherical (and hyperbolic) spaces or the publications [14,21] about Voronoi tessellations on the sphere. Let us also mention here the work [30], which studies intersection probabilities for deterministic and random cones, and [12], where random tessellations of the 2-dimensional sphere generated by a gravitational allocation scheme are investigated.…”