On-line packing sequences of cubes in the unit cube
type: Item
instance of: scholarly article
title: On-line packing sequences of cubes in the unit cube
zbMATH Open document ID: 0891.52009
DOI: 10.1023/A:1004953109743
author: Janusz Januszewski
author: Marek Lassak
published in: Geometriae Dedicata
publication date: 1998-06-12T00:00:00Z
review text: Meir and Moser proved that every sequence of \(d\)-dimensional cubes of total volume at most \(2(1/2)^d\) permits a translative packing in the unit cube of \(E^d\) [\textit{A. Meir} and \textit{L. Moser}, J. Comb. Theory 5, 126-134 (1968; Zbl 0165.25202)]. This estimate is the best possible, which follows from the example of two cubes of edge length \(1/2 +\varepsilon,\) where \(\varepsilon > 0\) is arbitrarily small (an equivalent observation is in \textit{J. W. Moon} and \textit{L. Moser} [Colloq. Math. 17, 103-110 (1967; Zbl 0152.39502)]). The authors' aim is to show that for \(d \geq 5\) the above property holds true also under the on-line restriction. Moreover, some estimates for dimensions smaller than 5 are given. It is proved that every sequence of squares of total area at most \(5/16\) can be on-line packed in the unit square, and that every sequence of cubes in \(E^d,\) where \(d\in \{3,4\},\) of total volume at most \((3/2)(1/2)^d\) can be on-line packed in the unit cube.
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 1098609
zbMATH Keywords: packing
zbMATH Keywords: on-line packing
zbMATH Keywords: cube
zbMATH Keywords: volume
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: a:1004953109743
OpenAlex ID: W104666939
description: scientific article; zbMATH DE number 1098609