Random points in halfspheres1. Issue 1 (8th February 2016) - Bárány, Imre; Hug, Daniel; Reitzner, Matthias; Schneider, Rolf
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