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A procedure for packing or covering a given convex bodyK with a sequence of convex bodies {C i} is anon-line method if the setC i are given in sequence. andC i+1 is presented only afterC  i  has been put in place, without the option of changing the placement afterward. The “one-line” idea was introduced by Lassak and Zhang [6] who found an asymptotic volume bound for the problem of on-line packing a cube with a sequence of convex bodies. In this note a problem of Lassak is solved, concerning on-line covering a cube with a sequence of cubes, by proving that every sequence of cubes in the Euclidean spaceE  d  whose total volume is greater than 4 d  admits an on-line covering of the unit cube. Without the “on-line” restriction, the best possible volume bound is known to be 2 d −1, obtained by Groemer [2] and, independently, by Bezdek and Bezdek [1]. The on-line covering method described in this note is based on a suitable cube-filling Peano curve.

On-line covering a cube by a sequence of cubes