Random convex hulls: Floating bodies and expectations (Q1318228)

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 \(n\).