Saturated packings and reduced coverings obtained by perturbing tilings

Let a normed space X possess a tiling T consisting of unit balls. We show that any packing P of X obtained by a small perturbation of T is completely translatively saturated; that is, one cannot replace finitely many elements of P by a larger number of unit balls such that the resulting arrangement is still a packing. trast with that, given a tiling T of R n with images of a convex body C under Euclidean isometries, there may exist packings P consisting of isometric images of C obtained from T by arbitrarily small perturbations which are no longer completely saturated. This means that there exists some positive integer k such that one can replace k − 1 members of P by k isometric copies of C without violating the packing property. However, we quantify a tradeoff between the size of the perturbation and the minimal k such that the above phenomenon occurs. ous results are obtained for coverings.