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

TL;DR

This paper studies the asymptotic behavior of convex cones generated by random points on half-spheres, establishing convergence of their geometric characteristics and linking them to Poisson point processes.

Contribution

It introduces a novel approach connecting random cones from half-spheres to Poisson point processes, providing explicit formulas for geometric functionals and answering prior open questions.

Findings

Weak convergence of the $f$-vector of the convex cone as n→∞

Explicit formulas for expected facets and intrinsic volumes

Convergence of expected Grassmann angles and conic intrinsic volumes

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 $n\to\infty$, 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 B\'ar\'any, 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 $\|x\|^{-(d+\gamma)}$, 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 $\gamma>0$.