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POISSON HYPERPLANE PROCESSES AND APPROXIMATION OF CONVEX BODIES
Author(s) -
Hug Daniel,
Schneider Rolf
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12040
Subject(s) - hyperplane , mathematics , polytope , combinatorics , intersection (aeronautics) , bounded function , regular polygon , convex polytope , poisson distribution , convex body , limit (mathematics) , mathematical analysis , convex set , convex hull , geometry , convex optimization , statistics , engineering , aerospace engineering
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.