, Volume 3, Issue 3, pp 387-393

Distribution-independent properties of the convex hull of random points

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Abstract

Denote byV n (d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure μ in Euclideand-spaceE d. Ifd=2 ord=3 and μ is the measure corresponding to the uniform distribution on a convex body inE d, Affentranger and Badertscher derived that $$V_{d + 2m}^{(d)} = \sum\limits_{k = 1}^m {(2^{2k} - 1)} \frac{{B_{2k} }}{k}\left( {\begin{array}{*{20}c} {d + 2m} \\ {2k - 1} \\ \end{array} } \right)V_{d + 2m - 2k + 1}^{(d)} \left( {m = 1,2, \ldots } \right)$$ where the constantsB 2k are the Bernoulli numbers. We verify this identity for arbitrary measures μ in spaces of arbitrary dimensiond. The extended version of the identity follows from a more general theorem, which is proved by constructing a moment functionalL such that $$V_{d + 1 + p} = \left( {\begin{array}{*{20}c} {d + 1 + p} \\ {d + 1} \\ \end{array} } \right)L\left[ {x^p + \left( {1 - x} \right)^p } \right]\left( {p = 0,1, \ldots } \right)$$