O ct 2 00 1 On the existence of completely saturated packings and completely reduced coverings

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....

On the Existence of Completely Saturated Packings and Completely Reduced Coverings

2 00 8 On densest packings of equal balls of R n and Marcinkiewicz spaces

We investigate, by “ à la Marcinkiewicz” techniques applied to the (asymptotic) density function, how dense systems of equal spheres of Rn, n ≥ 1, can be partitioned at infinity in order to allow the computation of their density as a true limit and not a limsup. The density of a packing of equal balls is the norm 1 of the characteristic function of the systems of balls in the sense of Marcinkie...

Notions of denseness

The notion of a completely saturated packing [1] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent dense packings and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, ...

On densest packings of equal balls of Rn and Marcinkiewicz spaces

We investigate, by “ à la Marcinkiewicz” techniques applied to the (asymptotic) density function, how dense systems of equal spheres of Rn, n ≥ 1, can be partitioned at infinity in order to allow the computation of their density as a true limit and not a limsup. The density of a packing of equal balls is the norm 1 of the characteristic function of the systems of balls in the sense of Marcinkie...