Journal of Theoretical Probability

, Volume 3, Issue 3, pp 387–393

Distribution-independent properties of the convex hull of random points

• Christian Buchta
Article

DOI: 10.1007/BF01061259

Buchta, C. J Theor Probab (1990) 3: 387. doi:10.1007/BF01061259

Abstract

Denote byVn(d) the expected volume of the convex hull ofn points chosen independently according to a given probability measure μ in Euclideand-spaceEd. Ifd=2 ord=3 and μ is the measure corresponding to the uniform distribution on a convex body inEd, 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 constantsB2k 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
$$V_{d + 1 + p} = \left( {\begin{array}{*{20}c} {d + 1 + p} \\ {d + 1} \\ \end{array} } \right)L\left[ {x^p + \left( {1 - x} \right)^p } \right]\left( {p = 0,1, \ldots } \right)$$

Key Words

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