Bárány, I.; Vitale, R. A. Random convex hulls: Floating bodies and expectations. (English) Zbl 0795.60008 J. Approximation Theory 75, No. 2, 130-135 (1993). Let \(K\) be a fixed compact, convex subset of \(R^ d\) with non-empty interior, and \(X_ 1,X_ 2, \dots\) independent random points uniformly distributed in \(K\). The expectation \(E{\mathbf X}_ n\) of the successive convex hulls \({\mathbf X}_ n\) is defined in terms of the support function. The function \(v(x)\) is defined for \(x \in K\) as the minimal volume of \(K \cap H\) for all halfspaces \(H\) which contain \(x\). For each \(\varepsilon>0\), \(K(\varepsilon)\) is the subset of \(K\) where \(v(x) \leq \varepsilon\). The associated floating body is \(K \backslash K (\varepsilon)\). The main result is the following approximation theorem: There are constants \(0<a<b<\infty\) such that \(K(a/n) \subset K \backslash E{\mathbf X}_ n \subset K(b/n)\) holds for all \(n\). Reviewer: R.Wegmann (Garching) Cited in 2 Documents MSC: 60D05 Geometric probability and stochastic geometry 52A22 Random convex sets and integral geometry (aspects of convex geometry) Keywords:random convex hulls; approximation theorem PDFBibTeX XMLCite \textit{I. Bárány} and \textit{R. A. Vitale}, J. Approx. Theory 75, No. 2, 130--135 (1993; Zbl 0795.60008) Full Text: DOI