On-line covering a cube by a sequence of cubes

Abstract

A procedure for packing or covering a given convex body K with a sequence of convex bodies {Ci} is an on-line method if the set Ci are given in sequence. and Ci+1 is presented only after Ci 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 space Ed whose total volume is greater than 4d admits an on-line covering of the unit cube. Without the "on-line" restriction, the best possible volume bound is known to be 2d -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. © 1994 Springer-Verlag New York Inc.