POISSON HYPERPLANE PROCESSES AND APPROXIMATION OF CONVEX BODIES. Issue 3 (4th May 2020)

• Record Type: Journal Article
• Authors: Hug, Daniel; Schneider, Rolf
• Is Part Of: Mathematika. Volume 66:Issue 3(2020)
• Page Start: 713
• Page End: 732
• Publication Date: 2020-05-04
• Subjects: 60D05 (primary) -- 52A27 (secondary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12040
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: 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.