|TEX2HTML_WRAP odzimierz Kuperburg, Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310, USA|
|Covering the cube with equal balls|
Define cd(k), the covering radius, as the minimum radius of k congruent balls that can cover the unit d-dimensional cube. (The packing radius is defined similarly.) In general, the problem of determining the values of cd(k) as well as the corresponding packing problem seem to be extremely difficult even in dimension 2. On the packing problem, several results have been obtained in dimension 2by various authors, and a few results in dimension 3, mainly by J. Schaer. We discuss the covering problem and we determine cd(2), c3(3), c3(4), c3(8), c4(4), and c4(16) along with the optimal configurations of balls that produce them. Also, we state conjectures on the remaining values of c3(k) and their ball configurations for .