2018/01/24 by Zakhar Kabluchko, Kabluchko, Zakhar, Alexander Marynych +5 · 1 citation
Computer Science · Mathematics · #52A22 #52B11 #60D05 (Primary) 52A55 #60F05 (Secondary) #FOS: Mathematics #Geochemistry and Geologic Mapping #Metric Geometry (math.MG) #Morphological variations and asymmetry #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1801.08008
openalex publication_date 2018/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let U1,U2,\… be random points sampled uniformly and independently\nfrom the d-dimensional upper half-sphere. We show that, as n\→\∞, the\nf-vector of the (d+1)-dimensional convex cone Cn generated by\nU1,\…,Un weakly converges to a certain limiting random vector, without\nany normalization. We also show convergence of all moments of the f-vector of\nCn and identify the limiting constants for the expectations. We prove that\nthe expected Grassmann angles of Cn can be expressed through the expected\nf-vector. This yields convergence of expected Grassmann angles and conic\nintrinsic volumes and answers thereby a question of B 'ar 'any, Hug, Reitzner\nand Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our\napproach is based on the observation that the random cone Cn weakly\nconverges, after a suitable rescaling, to a random cone whose intersection with\nthe tangent hyperplane of the half-sphere at its north pole is the convex hull\nof the Poisson point process with power-law intensity function proportional to\n\‖x\‖-(d+\γ), where \γ=1. We compute the expected number of\nfacets, the expected intrinsic volumes and the expected T-functional of this\nrandom convex hull for arbitrary \γ>0.\n