Coating by cubes
type: Item
instance of: scholarly article
title: Coating by cubes
zbMATH Open document ID: 0814.52008
author: Tamás Hausel
author: Károly Bezdek
published in: Beiträge zur Algebra und Geometrie
publication date: 1994-12-12T00:00:00Z
full work available at URL: https://eudml.org/doc/225588
full work available at URL: no.1
review text: Consider \(P_ 0,P_ 1,\dots,P_ n\) convex \(d\)-polytopes in \(\mathbb{E}^ d\) \((d \geq 2)\) with pairwise disjoint interiors. Then \(P_ 0\) is said to be coated by \(P_ 1,\dots,P_ n\) if \(P_ 0 \subset \text{int}(\bigcup^ n_{i = 0} P_ i)\), where \(\text{int}()\) denotes the interior of a set. The authors are interested in the minimal number of convex polytopes that coat a given convex polytope. More precisely they investigate the following two problems: \(\bullet\) What is the minimum number of \(d\)-simplices that can coat a \(d\)-simplex in \(\mathbb{E}^ d\)? \(\bullet\) Is \(2^{d - 1} + 2\) the minimum number of \(d\)-cubes that can coat a \(d\)-cube? Using elementary methods they prove the following two results in this direction: \(\bullet\) Every \(d\)-simplex can be coated by \((2d - 1)\) \(d\)-simplices in \(\mathbb{E}^ d\). \(\bullet\) The minimum number of translates of a \(d\)-cube that can coat a given \(d\)-cube in \(\mathbb{E}^ d\) is at least \(2^ d\). If all \(d\)-cubes are translates of each other, then \(2^ d\) can be replaced by \(2^{d + 1} - 2\).
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 702462
zbMATH Keywords: convex \(d\)-polytopes
zbMATH Keywords: coated
zbMATH Keywords: \(d\)-simplex
zbMATH Keywords: \(d\)-cube
reviewed by: Charles Leytem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 702462