A new approach to weak convergence of random cones and polytopes

Abstract

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation generated by n independent and uniformly distributed random linear hyperplanes in R d+1 weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in R d , as n → ∞.

Faces in random great hypersphere tessellations

“…Using (29) with replaced by d it is easy to check that the condition is satisfied for the value of γ given in Theorem 5.13. Thus, in the large n limit, the typical and the weighted spherical cells "look like" the corresponding Euclidean ones (with the above choice of γ) divided by n. In fact, it is possible to state and prove such results rigorously, see [20]. Now, consider a small spherical polytope P n (for example, Z n,d or W n,d ) which is close to 1 n P , where P is fixed Euclidean polytope.…”

“…Remark 2.2. Taking k = d in the previous definition we get back the spherical random polytope whose conical version was studied in [16,20] under the name Schäfli random cone.…”

“…, d} we introduce two different types of random k-dimensional spherical faces associated with a great hypersphere tessellation, the k-dimensional typical spherical face and the k-dimensional weighted typical spherical face. In the equivalent language of conical random tessellations, the typical spherical kface generalizes to lower dimensions the concept of the Schläfli random cone studied in [8,16,20]. We investigate the relation between and provide probabilistic interpretations of these lower-dimensional spherical random polytopes.…”