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Random points in halfspheres
Author(s) -
Bárány Imre,
Hug Daniel,
Reitzner Matthias,
Schneider Rolf
Publication year - 2017
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20644
Subject(s) - convex hull , combinatorics , mathematics , polytope , convex polytope , convex body , regular polygon , convex set , euclidean space , vertex (graph theory) , geometry , graph , convex optimization
A random spherical polytope P n in a spherically convex set K ⊂ S das 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