×

Random convex hulls: Floating bodies and expectations. (English) Zbl 0795.60008

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 \(n\).

MSC:

60D05 Geometric probability and stochastic geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
PDFBibTeX XMLCite
Full Text: DOI