Random points in halfspheres |
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| Authors: | Imre Bárány Daniel Hug Matthias Reitzner Rolf Schneider |
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| Affiliation: | 1. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary;2. Department of Mathematics, University College London, London, England;3. Karlsruhe Institute of Technology, Department of Mathematics, Karlsruhe, Germany;4. Department of Mathematics, University of Osnabrück, Osnabrück, Germany;5. Department of Mathematics, Albert‐Ludwigs‐Universit?t, Freiburg i. Br., Germany |
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| Abstract: | A random spherical polytope Pn in a spherically convex set  as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of Pn 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 Pn, 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 |
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| Keywords: | spherical spaces random polytopes in halfspheres face numbers (spherical) surface area volume and mean width Hausdorff distance |
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as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of Pn 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 Pn, 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