On the limit of multiple packings

For a collection of subsets of a setS, letnk be the maximal number of sets from such that no point ofS belongs to more thank of them. Then the limitδ of the sequencenk/k is related to the valuev of the game in which the minimizer selects a setα in, the maximizer a pointx inS, and the payoff isϰα(x), byδ≤v−1≤infvδv. Herev is any finite subset ofS and δv the corresponding limit. Analogous results hold for covering.The bounds obtained in this way fornk and in particularn1 reflect the influence of the boundary ofS when, for instance,S is a convex body to be packed with congruent copies of another body. In some cases, the bounds are sharp.