Wlodzimierz Kuperberg

Packing and Covering with Convex Sets

Publisher Summary This chapter describes packing and covering with convex sets and discusses arrangements of sets in a space E, which should have a structure admitting the notions of congruence, measure, and convexity. Given a domain in E, a packing in the domain is an arrangement the members of which are all contained in the domain and have mutually disjoint interiors, and a covering of the domain is an arrangement whose union contains the domain. A packing in or a covering of the whole space E is called a packing or a covering, respectively. An arrangement that is a packing and a covering at the same time is called a tiling. All the known proofs of the theorem of Minkowski–Hlawka and its refinements are nonconstructive. The concepts of packing and covering are well defined in hyperbolic geometry. While high-density packings and low-density coverings can be considered efficient, the definition of density allows some undesired local deviations to occur that go contrary to the intuitive concept of efficiency.

Double-lattice packings of convex bodies in the plane

Mahler [7] and Fejes Tóth [2] proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least √3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Tóth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of−K. The lower bound of √3/2 for packing densities produced here is an improvement of the bounds obtained previously in [5] and [6].

Maximum density space packing with congruent circular cylinders of infinite length

We determine what is the maximum possible (by volume) portion of the three-dimensional Euclidean space that can be occupied by a family of non-overlapping congruent circular cylinders of infinite length in both directions. We show that the ratio of that portion to the whole of the space cannot exceed π/√12 and it attains π/√12 when all cylinders are parallel to each other and each of them touches six others. In the terminology of the theory of packings and coverings, we prove that the space packing density of the cylinder equals π/√12, the same as the plane packing density of the circular disk.

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.

A Survey of Recent Results in the Theory of Packing and Covering

The theory of packing and covering, originated as an offspring of number theory and crystallography early in this century, has quickly gained interest of its own and is now an essential part of discrete geometry. The theory owes its early development to its aesthetic appeal and its classical flavor, but more recently, some of its topics have been found related to the rapidly developing areas of mathematics connected with computer science, and the theory of packing and covering has been boosted by a renewed interest.

Knotted lattice-like space fillers

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On-line covering a cube by a sequence of cubes

\mathbb{E}^3

Maximum density space packing with parallel strings of spheres

by a lattice. More generally, a lattice-like space filler is constructed that is a handlebody of arbitrary genus, whose handles are arbitrarily knotted and arbitrarily linked with each other, and, furthermore, through which mutually disjoint tunnels have been drilled, knotted, and linked with each other arbitrarily.