Separation in totally-sewn 4-polytopes with the decreasing universal edge property
type: Item
instance of: scholarly article
title: Separation in totally-sewn 4-polytopes with the decreasing universal edge property
zbMATH Open document ID: 1238.52004
published in: Beiträge zur Algebra und Geometrie
publication date: 2012-02-29T00:00:00Z
review text: The separation number \(s(O,P)\) of a \(d\)-polytope \(P\) with respect to a point \(O\) in the interior of \(P\) is the minimum number of hyperplanes necessary to strictly separate \(O\) from any facet of \(P\). According to a variation of the Gohberg-Markus-Hadwiger conjecture that is known as the Bezdek's separation conjecture, \(s(O,P)\leq 2^d\) for all \(d\)-polytopes. In general this is open but is proved for cyclic \(d\)-polytopes, see [\textit{K. Bezdek} and \textit{T. Bisztriczky}, Geom. Dedicata 68, No. 1, 29--41 (1997; Zbl 0965.52014)]. The authors verify the conjecture for totally-sewn 4-polytopes with the property that the number of universal edges does not increase as the polytopes are constructed. Here a polytope is called totally-sewn if it can be constructed by iterations of the following procedure that starts with a 4-simplex: the next neighbourly 4-polytope is the convex hull of the previous polytope \(P\) and a suitably located exterior point of \(P\).
reviewed by: Victor Alexandrov
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52B12
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A37
zbMATH DE Number: 6010041
zbMATH Keywords: Gohberg-Markus-Hadwiger covering number
zbMATH Keywords: neighbourly polytope
zbMATH Keywords: separation problem
zbMATH Keywords: totally-sewn polytope
author: T. Bisztriczky
author: Ferenc Fodor
author: Deborah Oliveros
MaRDI profile type: Publication
full work available at URL: s13366-011-0069-z
OpenAlex ID: W2317116406
cites work: The problem of illumination of the boundary of a convex body by affine subspaces
cites work: Hadwiger's covering conjecture and low dimensional dual cyclic polytopes
cites work: A proof of Hadwiger's covering conjecture for dual cyclic polytopes
cites work: Separation in neighbourly 4-polytopes
cites work: Q4464708
cites work: Excursions into combinatorial geometry
cites work: Separation in semi-cyclic 4-polytopes
cites work: Q4434671
cites work: Combinatorial problems on the illumination of convex bodies
cites work: Neighborly polytopes
cites work: How many cyclic subpolytopes can a non-cyclic polytope have?
DOI: 10.1007/S13366-011-0069-Z
description: scientific article; zbMATH DE number 6010041