On-line covering a box by cubes (Q1892790)

The paper considers on-line coverings [the author and \textit{J. Zhang}, Discrete Comput. Geom. 6, No. 1, 1-7 (1991; Zbl 0727.52004)] of the box \(B(s_ 1, \ldots, s_ d)= \{(x_ 1, \ldots, x_ d)\): \(0\leq x_ j\leq s_ j\), where \(j=1, \ldots, d\}\) in \(E^ d\). The author uses two methods to realize the on-line covering of \(B(s_ 1, \ldots, s_ d)\) by cubes of sides at most 1 and gives an estimate (Theorem 1) for the total volume of the cubes. Then it is shown in Theorem 2 that every sequence of \(d\)-dimensional cubes of sides at most 1 whose total volume is at least a number (given in Theorem 2) permits an on-line covering for a cube of sides \(s\) in \(E^ d\). Asymptotically this total volume is \(s^ d\).