Cones generated by random points on half-spheres and convex hulls of Poisson point processes - Kabluchko, Zakhar; Marynych, Alexander; Temesvari, Daniel; Thäle, Christoph (Probability theory and related fields, 2019)
Let U 1 , U 2 , … be random points sampled uniformly and independently from the d -dimensional upper half-sphere. We show that, as n → ∞ , the f -vector of the ( d + 1 ) -dimensional convex cone C n generated by U 1 , … , 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 ‖ 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 .