The expected value of some functions of the convex hull of a random set of points sampled inR
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Authors: | Isaac Meilijson |
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Institution: | (1) Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, 69978 Ramat Aviv, Tel Aviv, Israel |
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Abstract: | This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the
convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton
(d−1)/(d+1), which generalizes Rényi and Sulanke’s asymptotic raten
(1/3) ford=2 and agrees with Raynaud’s asymptotic raten
(d−1)/(d+1) for the expected number of facets, as it should be, by Bárány’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown
convex subset ofR
d
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Keywords: | |
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