Saturated packings and reduced coverings obtained by perturbing tilings

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