Random points in halfspheres1. Issue 1 (8th February 2016)

• Record Type: Journal Article
• Authors: Bárány, Imre; Hug, Daniel; Reitzner, Matthias; Schneider, Rolf
• Is Part Of: Random structures & algorithms. Volume 50:Issue 1(2017)
• Page Start: 3
• Page End: 22
• Publication Date: 2016-02-08
• Subjects: spherical spaces -- random polytopes in halfspheres -- face numbers -- (spherical) surface area, volume and mean width -- Hausdorff distance; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20644
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: A random spherical polytope P n in a spherically convex set K ⊂ S d as considered here is the spherical convex hull of n independent, uniformly distributed random points in K . The behaviour of P n for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when K is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as n tends to infinity, of the expectation of several characteristics of P n, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 3–22, 2017