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

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 normaliz...

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Bibliographic Details
Published in:	Probability theory and related fields Vol. 175; no. 3-4; pp. 1021 - 1061
Main Authors:	Kabluchko, Zakhar, Marynych, Alexander, Temesvari, Daniel, Thäle, Christoph
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2019 Springer Nature B.V
Subjects:	Algorithms Computational geometry Cones Constraining Convergence Convexity Economics Finance Hulls Hyperplanes Insurance Management Mathematical and Computational Biology Mathematical and Computational Physics Mathematics Mathematics and Statistics North Pole Operations Research/Decision Theory Probability Probability Theory and Stochastic Processes Quantitative Finance Rescaling Statistics for Business Theoretical 60D05 Primary 52A22 52B11 Conic intrinsic volume Blaschke–Petkantschin formula Convex cone Spherical integral geometry Convex hull Poisson point process Secondary 52A55 Random polytope Vector 60F05
ISSN:	0178-8051, 1432-2064
Online Access:	Get full text
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Summary:	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 .
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0178-8051 1432-2064
DOI:	10.1007/s00440-019-00907-3