The expected volume of a random polytope in a ball* (original) (raw)

Journal of Microscopy, 1988

Abstract

For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d)n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d)n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d)n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases.

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