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

Abstract

Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $n\rightarrow \infty$, the f-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots ,U_n$ 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 et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644). 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 $\gamma >0$.