A sufficient condition for the covering of the unit cube by homothetic copies in the \(n\)-dimensional Euclidean space
type: Item
instance of: scholarly article
title: A sufficient condition for the covering of the unit cube by homothetic copies in the \(n\)-dimensional Euclidean space
zbMATH Open document ID: 0556.52008
author: Károly Bezdek
author: A. Bezdek
published in: Beiträge zur Algebra und Geometrie
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138261
review text: L. Fejes Tóth, independently of \textit{D. J. Newman} [SIAM Rev. 24, 77 (1982), \url{http://dx.doi.org/10.1137/1024007}] phrased the following problem: Let \(K\) be a convex body of volume 1 in the \(n\)-dimensional Euclidean space \(E^ n\). Denote by \(f_ n(K)\) the smallest positive number which has the following property. Any collection of bodies, which are homothetic to \(K\) with total volume \(f_ n(K)\) can cover \(K\). The problem is to determine \(f_ n(K)\). L. Fejes Tóth conjectured that \(2\leq f_ 2(K)\leq 3\) and \(f_ n(C)=2^ n-1,\) where \(C\) denotes the unit hypercube in \(E^ n\). The paper contains the proofs of the statements \(f_ n(C)=2^ n-1,\) \(2\leq f_ 2(K)<12\) and discusses some remarks about similar questions. Independently of the authors \textit{A. Meir} [SIAM Rev. 25, No. 1, 99--101 (1983), \url{http://dx.doi.org/10.1137/1025012}] published a proof for \(f_ n(C)=2^ n-1\).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 3886660
zbMATH Keywords: homothetic covering
zbMATH Keywords: hypercube
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3886660