On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls

On Finite Lattice Coverings

We consider nite lattice coverings of strictly convex bodies K. For planar centrally symmetric K we characterize the nite arrangements Cn such that conv Cn Cn + K, where Cn is a subset of a covering lattice for K (which satisses some natural conditions). We prove that for a xed lattice the optimal arrangement (measured with the parametric density) is either a sausage, a socalled double sausage ...

On 2-Coverings and 2-Packings of Laminar Families

Let H be a laminar family of subsets of a groundset V A k cover of H is a multiset C of edges on V such that for every subset S in H C has at least k edges that have exactly one end in S A k packing of H is a multiset P of edges on V such that for every subset S in H P has at most k u S edges that have exactly one end in S Here u assigns an integer capacity to each subset in H Our main results ...

On sequences of lattice packings

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 packing P with congruent replicas of a body K is n-saturated if no n− 1 members of it can be replaced with n replicas of K, and it is completely saturated if it is n-saturated for each n ≥ 1....

Finite Sphere Packings and Sphere Coverings