On the zone of the boundary of a convex body

2013/06/10 by Orit E. Raz, Orit Esther Raz, Raz, Orit Esther
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #Point processes and geometric inequalities #cs.CG

paper · pdf · doi:10.48550/arxiv.1306.2104

arxiv created 2013/06/10 · openalex publication_date 2013/06/10 · arxiv updated 2013/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an arrangement \A of n hyperplanes in \R^d and the zone \Z in \A of the boundary of an arbitrary convex set in \R^d in such an arrangement. We show that, whereas the combinatorial complexity of \Z is known only to be O<n^d-1log n> \citeAPS, the outer part of the zone has complexity O<n^d-1> (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for d=2).

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