Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Abstract

Let $U_1,U_2,\dots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to \infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\dots ,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\Vert x{\Vert }^{-(d+\gamma )}$, where $\gamma =1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $\gamma >0$.

31 pages, 2 figures