Some basic properties of packing and covering constants

Abstract

This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collection K of convex bodies. There are two possibilities to construct such packings and coverings: One may permit that the convex bodies from K are used repeatedly, or one may require that these bodies should be used at most once. In each case one can define the packing and covering constants of K as, respectively, the least upper bound and the greatest lower bound of the densities of all such packings and coverings. Three theorems are proved. First it is shown that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants. Then, a quantitative continuity theorem is proved which shows in particular that the packing and covering constants depend, in a certain sense, continuously on K. Finally, a kind of a transference theorem is proved, which enables one to evaluate the packing and covering constants when no repetitions are allowed from the case when repetitions are permitted. Furthermore, various consequences of these theorems are discussed. © 1986 Springer-Verlag New York Inc.