Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non−1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn≥1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn−1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn≥1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onK⊂Ed (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofEd with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.