The expected value of some functions of the convex hull of a random set of points sampled in R d

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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 of n points taken from the uniform distribution on a d-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional to n (d-1)/(d+1), which generalizes Rényi and Sulanke's asymptotic rate n (1/3) for d=2 and agrees with Raynaud's asymptotic rate n (d-1)/(d+1) for the expected number of facets, as it should be, by Bárány's result that the expected number of s-dimensional faces has order of magnitude independent of s. Our formulas agree with the ones Efron obtained for d=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset of R d .

Original languageEnglish
Pages (from-to)341-352
Number of pages12
JournalIsrael Journal of Mathematics
Volume72
Issue number3
DOIs
StatePublished - Oct 1990

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