A. Bezdek

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.

On convex polygons of maximal width

Abstract. In this paper we consider the problem of finding the n-sided (

On a Generalization of Tarski's Plank Problem

n\geq 3

Facets with fewest vertices

) polygons of diameter 1 which have the largest possible width wn. We prove that

On the density of dual circle coverings

w_4=w_3= {\sqrt 3 \over 2}

Covering an Annulus by Strips

and, in general,

Finite and uniform stability of sphere coverings

w_n \leq \cos {\pi \over 2n}

On illumination in the plane by line segments

. Equality holds if n has an odd divisor greater than 1 and in this case a polygon

Interior points of the convex hull of few points in\(\mathbb{E}^d \)

\cal P

On the second smallest distance between finitely many points on the sphere

is extremal if and only if it has equal sides and it is inscribed in a Reuleaux polygon of constant width 1, such that the vertices of the Reuleaux polygon are also vertices of