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

• 成果类型: Article
• 署名作者: Kabluchko, Zakhar; Marynych, Alexander; Temesvari, Daniel; Thaele, Christoph
• 署名单位: University of Munster; Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv; Ruhr University Bochum
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00907-3
• 发表日期: 2019
• 页码: 1021-1061
• 关键词: intrinsic volumes; probability

摘要

Let U1, U2,... be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We showthat, as n.8, the f -vector of the (d+1)-dimensional convex cone Cn generated by U1,..., Un weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f -vector of Cn and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of Cn 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 Barany 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 Cn weakly converges, after a suitable rescaling, to a random cone whose intersectionwith 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+.), where. = 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.