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

• Main Authors: Kabluchko, Zakhar, Marynych, Alexander, Temesvari, Daniel, Thaele, Christoph
• Format: Report
• Language: unknown
• Published: arXiv 2018
• Subjects: Probability math.PR; Metric Geometry math.MG; FOS Mathematics; 52A22, 60D05 Primary 52A55, 52B11, 60F05 Secondary; North Pole
• Online Access: https://dx.doi.org/10.48550/arxiv.1801.08008; https://arxiv.org/abs/1801.08008

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 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 $\|x\|^{-(d+γ)}$, where $γ=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $γ>0$. : 31 pages, 2 figures