On Grünbaum's problem about inner illumination of convex bodies
type: Item
instance of: scholarly article
title: On Grünbaum's problem about inner illumination of convex bodies
zbMATH Open document ID: 0861.52001
DOI: 10.1007/BF01874604
published in: Acta Mathematica Hungarica
publication date: 1996-05-13T00:00:00Z
review text: By an illumination set of a \(d\)-dimensional convex body \(K\) there is understood a subset \(F \subset bdK\) such that for every \(x\in bdK\) there exists a \(y\neq x\) in \(F\) such that the open interval \(]x,y[\) is contained in \(\text{int} K\). \(F\) is called primitive if it does not contain properly another illumination set. Grünbaum conjectured that a primitive illumination set contains at most \(2^d\) points. The author proves the conjecture for \(d=3\), equality holding only for \(K\) being combinatorially equivalent to a cube.
Mathematics Subject Classification ID: 52A15
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 878083
zbMATH Keywords: convex body
zbMATH Keywords: illumination
author: Valeriu Soltan
reviewed by: Günter Ewald
MaRDI profile type: Publication
cites work: Fixing systems and inner illumination
cites work: Q5547252
cites work: Q5617498
cites work: Q5582184
description: scientific article; zbMATH DE number 878083