Random polytopes and the Efron-Stein jackknife inequality.
For some \(d\geq2\), let \({\mathcal K}^2_+\) be the set of compact convex sets in \(d\)-dimensional Euclidean space with non-empty interior, positive Gaussian curvature and with boundary of differentiability class \({\mathcal C}^2\). For a fixed \(K\in{\mathcal K}^2_+\), choose points \(X_1,\ldots,X_n\) independently and uniformly at random. This paper is concerned with the number of vertices, \(N(K_n)\), and volume, \(V(K_n)\), of the convex hull \(K_n\) of the \(n\) points. Estimates are obtained for the variances of \(N(K_n)\) and \(V(K_n)\), from which strong laws of large numbers for these random variables follow. The essential step is the use of the jacknife inequality of \textit{B.~Efron} and \textit{C.~Stein} [Ann. Stat. 9, 586-596 (1981; Zbl 0481.62035)]. Results of \textit{I.~Bárány} [in: Second international conference in stochastic geometry, convex bodies and empirical measures. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 50, 43--50 (1997; Zbl 0890.52006)] concerning approximation of convex bodies by random polytopes and by best-approximating inscribed polytopes are strengthened. Analogous results for random polytopes with vertices on the boundary of the convex body are given.
scientific article; zbMATH DE number 2077591
scientific article