On-line packing sequences of cubes in the unit cube
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.
scientific article; zbMATH DE number 1098609