Intrinsic volumes and f-vectors of random polytopes
Let K be a convex body with interior points in \({\mathbb{R}}^ d\) and \(K_ n\) a random polytope in K, defined as the convex hull of n independent uniform random points in K. For \(\epsilon >0\), let \(K[\epsilon]:=\{x\in K:\) vol(K\(\cap H)\geq \epsilon\) for every halfspace H with \(x\in H\}\), which is a convex body for small \(\epsilon\). For certain functionals \(\phi\) on convex bodies, the asymptotic behaviour, for \(n\to \infty\), of the expectation of \(\phi (K_ n)\) compares well with the asymptotic behaviour of \(\phi\) (K[1/n]). For the volume, this fact was investigated and exploited in a paper by the author and \textit{D. G. Larman} [Mathematika 35, No.2, 274-291 (1988; Zbl 0674.52003)]. Here this powerful method is developed further. Among the main results are precise formulations of the following facts, here expressed in a loose way: (1) the expectation of \(V_ s(K)-V_ s(K_ n)\) is about \(V_ s(K)- V_ s(K[1/n])\), where \(V_ s\) denotes the s th intrinsic volume, (2) the expectation of the number of s-dimensional faces of \(K_ n\) is about \(n vol(K\setminus K[1/n])/vol K.\) Under special assumptions on K, the behaviour of K[1/n] for \(n\to \infty\) is known, which leads to more explicit results. For example, if K is a polytope, then E \(f_ s(K_ n)\sim (\log n)^{d-1}\), and if K is of class \(C^ 2\), then E \(f_ s(K_ n)\sim n^{(d-1)/(d+1)}\). Here \(s\in \{0,...,d-1\}\), \(f_ s\) is the number of s-faces, E denotes expectation, and g(n)\(\sim h(n)\) means that there are positive numbers \(c_ 1\), \(c_ 2\) independent of n such that \(c_ 1g(n)<h(n)<c_ 2g(n)\) for n large enough. Another result says that, for K of class \(C^ 2\) with positive Gauss curvature, the Hausdorff distance \(\delta\) satisfies E \(\delta\) (K,K\({}_ n)\sim (n^{-1}\log n)^{2/(d+1)}\).
scientific article; zbMATH DE number 4140699
scientific article