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Distribution-independent properties of the convex hull of random points

Authors:

Christian Buchta

Affiliation:

1. Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universit?t, Wiedner Hauptstrasse 8-10, A-1040, Vienna, Austria

Abstract:

Denote byV_n^(d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure mgr

in Euclideand-spaceE^d. Ifd=2 ord=3 and mgr

is the measure corresponding to the uniform distribution on a convex body inE^d, Affentranger and Badertscher derived that

$V_{d + 2m}^{(d)} = sumlimits_{k = 1}^m {(2^{2k} - 1)} frac{{B_{2k} }}{k}left( {begin{array}{*{20}c} {d + 2m} {2k - 1} end{array} } right)V_{d + 2m - 2k + 1}^{(d)} left( {m = 1,2, ldots } right)$

Keywords:	Random points random polytope convex hull geometric probability distribution-independent properties moment functional Bernoulli numbers
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