On-line covering a cube by a sequence of cubes (Q1330886)

According to the ``on-line'' idea by \textit{M. Lassak} and \textit{J. Zhang} [ibid. 6, No. 1, 1-7 (1991; Zbl 0727.52004)] a procedure for packing or covering a given convex body \(K\) with a sequence of convex bodies \(\{C_ i\}\) is called an on-line packing or on-line covering if the sets \(C_ i\) are given in sequence, and \(C_{i + 1}\) is presented only after \(C_ i\), one at a time, each of them must be assigned its place before the next one appears, and that the placement cannot be modified afterward. The author proves the following theorem: Every sequence of cubes in the Euclidean space \(E^ d\) whose sum of volumes is greater than \(4^ d\) admits an on-line covering of the unit cube. He describes the on-line method on the base of a suitable cube-filling Peano curve.