G. Fejes Tóth

New Results in the Theory of Packing and Covering

Let J be a system of sets. If all members of J are contained in a given set C and each point of C belongs to at most one member of J then J is said to be a packing into C. If, on the other hand, each point of C belongs to at least one member of J then we say that J is a covering of C. Packings and coverings have been considered in various spaces and on various combinatorial structures. Here we are interested in problems concerning packings and coverings consisting of convex bodies in spaces of constant curvature, i.e. in Euclidean, spherical and hyperbolic space. Instead of saying that J is a packing into the whole space or J is a covering of the whole space we shall simply use the terms J is a packing and J is a covering.

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.

CONVEXLY INDEPENDENT SETS

A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family ℱ of mutually disjoint compact convex sets such that any subfamily of at mostk members of ℱ is convexly independent, but no subfamily of sizen is.

Covering the plane with two kinds of circles

A sharp lower bound is given for the density of a covering of the plane with two kinds of circles

MULTIPLE PACKING AND COVERING OF SPHERES

A system of equal spheres is said to form a k-fold packing if each point of the space belongs to at most k spheres. Analogously, a system of equal spheres is said to form a k-foM covering if each point of the space belongs to at least k spheres. Let f~, be the supremum of tile densities of all k-fold packings of equal spheres in Euclidean n-space. Similarly, let A~ be the infimum of the densities of all k-fold coverings of the Euclidean n-space with equal spheres. Obviously, we have the trivial bounds f~ l/n as well as for small values of k and n. In his survey on multiple packing of spheres held on the Colloquium on Convexity in Copenhagen in 1964, L. FEW [7] posed the problem to find a non-trivial upper bound for gig valid for all values of k and n. As far as I know, no non-trivial lower bound for A~ was known for n>2 and k>l. A. FLORIAN [10] proved that

Nine convex sets determine a pentagon with convex sets as vertices

It is proved that if ℱ is a family of nine pairwise disjoint compact convex sets in the plane such that no member of ℱ is contained in the convex hull of the union of two other sets of ℱ, then ℱ has a subfamily ℱ′ with five elements such that no member of ℱ′ is contained in the convex hull of the union of the other sets of ℱ′.

Finite sphere packing and sphere covering

A basic problem of finite packing and covering is to determine, for a given number ofk unit balls in Euclideand-spaceEd, (1) the minimal volume of all convex bodies into which thek balls can be packed and (2) the maximal volume of all convex bodies which can be covered by thek balls. In the sausage conjectures by L. Fejes Tóth and J. M. Wills it is conjectured that, for alld≥5, linear arrangements of thek balls are best possible. In the paper several partial results are given to support both conjectures. Furthermore, some relations between finite and infinite (space) packing and covering are investigated.

Finite coverings by translates of centrally symmetric convex domains

Bambah and Rogers proved that the area of a convex domain in the plane which can be covered byn translates of a given centrally symmetric convex domainC is at most (n−1)h(C)+a(C), whereh(C) denotes the area of the largest hexagon contained inC anda(C) stands for the area ofC. An improvement with a term of magnitude √n is given here. Our estimate implies that ifC is not a parallelogram, then any covering of any convex domain by at least 26 translates ofC is less economic than the thinnest covering of the whole plane by translates ofC.

Densest packings of typical convex sets are not lattice-like

We show that ifP is a convex polygon which has no parallel sides, then the densest packing of the plane with congruent copies ofP is not lattice-like. As a corollary we obtain that, in the sense of Baire categories, for most convex disks densest packing is not lattice-like.

A geometrical analogue of the phase transformation of crystals

Abstract A geometrical problem is discussed whose solution shows a close analogy to the transformation of one allotropic crystalline form to another.