Károly Bezdek

Classical topics in discrete geometry

Classical Topics Revisited.- Sphere Packings.- Finite Packings by Translates of Convex Bodies.- Coverings by Homothetic Bodies - Illumination and Related Topics.- Coverings by Planks and Cylinders.- On the Volume of Finite Arrangements of Spheres.- Ball-Polyhedra as Intersections of Congruent Balls.- Selected Proofs.- Selected Proofs on Sphere Packings.- Selected Proofs on Finite Packings of Translates of Convex Bodies.- Selected Proofs on Illumination and Related Topics.- Selected Proofs on Coverings by Planks and Cylinders.- Selected Proofs on the Kneser-Poulsen Conjecture.- Selected Proofs on Ball-Polyhedra.

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

Finite and Uniform Stability of Sphere Packings

Abstract. The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε , is the identity rearrangement. For example, the lattice packings Dd and Ad for d ≥ 3 in Ed are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in Ed for d ≥ 2 is not uniformly stable. A packing of balls is called finitely stable if any finite subfamily of the packing is fixed by its neighbors. If a packing is uniformly stable, then it is finitely stable. On the other hand, the cubic lattice packings mentioned above, which are not uniformly stable, are nevertheless finitely stable.

Hadwiger-Levi’s Covering Problem Revisited

A well-known problem of discrete geometry is due to Hadwiger (1957), (1960) and Levi (1955). The following conjecture concerning this problem was published by Hadwiger (1957), (1960) and also by Gohberg and Markus (1960): Any convex body of E d , d ≥ 1 (i.e. any compact convex subset of the d-dimensional Euclidean space E d with non-empty interior) can be covered by 2 d smaller homothetic bodies and equality is attained only for d-dimensional parallelotopes. This conjecture has stimulated a lot of research in geometry. To survey the basic results we need some simple definitions.

The problem of illumination of the boundary of a convex body by affine subspaces

The main result of this paper is the following theorem. If P is a convex polytope of E d with affine symmetry, then P can be illuminated by eight (d - 3)-dimensional affine subspaces (two (d- 2)-dimensional affine subspaces, resp.) lying outside P, where d 3= 3. For d = 3 this proves Hadwigers conjecture for symmetric convex polyhedra namely, it shows that any convex polyhedron with affine symmetry can be covered by eight smaller homothetic polyhedra. The cornerstone of the proof is a general separation method.

Lectures on Sphere Arrangements – the Discrete Geometric Side

1. Unit Sphere Packings.- 2. Proofs on Unit Sphere Packings.- 3. Contractions of Sphere Arrangements.- 4. Proofs on Contractions of Sphere Arrangements.- 5. Ball-Polyhedra and Spindle Convex Bodies.- 6. Proofs on Ball-Polyhedra and Spindle Convex Bodies.- 7. Coverings by Cylinders.- 8. Proofs on Coverings by Cylinders.- 9. Research Problems - an Overview.- Glossary.- References.

Rigidity of ball-polyhedra in Euclidean 3-space

In this paper we introduce ball-polyhedra as finite intersections of congruent balls in Euclidean 3-space. We define their duals and study their face-lattices. Our main result is an analogue of Cauchys rigidity theorem.

The Kneser---Poulsen Conjecture for Spherical Polytopes

Abstract If a finite set of balls of radius π/2 (hemispheres) in the unit sphere Sn is rearranged so that the distance between each pair of centers does not decrease, then the (spherical) volume of the intersection does not increase, and the (spherical) volume of the union does not decrease. This result is a spherical analog to a conjecture by Kneser (1954) and Poulsen (1955) in the case when the radii are all equal to π/2.

A Proof of Hadwiger's Covering Conjecture for Dual Cyclic Polytopes

In 1957, H. Hadwiger conjectured that a convex body K in a Euclidean d-space, d ≥ 1, can always be convered by 2d smaller homotheti copies of K. We verify this conjecture when K is the polar of a cyclic d-polytope.

On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

A Minkowski space Md=(Rd, ??) is just Rd with distances measured using a norm ??. A norm ?? is completely determined by its unit ball {x?Rd??x??1} which is a centrally symmetric convex body of the d-dimensional Euclidean space Ed. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in Md, d?3. In fact, we deal with a somewhat more general problem namely, we give upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in Ed, d?3.