POISSON HYPERPLANE PROCESSES AND APPROXIMATION OF CONVEX BODIES

A natural model for the approximation of a convex body K in R d by random polytopes is obtained as follows. Take a stationary Poisson hyperplane process in R d , and consider the random polytope Z K defined as the intersection of all closed halfspaces containing K that are bounded by hyperplanes of the process not intersecting K . If f is a functional on convex bodies, then for increasing intensities of the process, the expectation of the difference f ( Z K ) − f ( K )may or may not converge to zero. If it does, then the order of convergence and possible limit relations are of interest. We study these questions if f is either the mean width or the hitting functional. As a consequence, some results on the facet number are obtained.