Discrete & Computational Geometry

, Volume 20, Issue 1, pp 111-130

First online:

Finite and Uniform Stability of Sphere Packings

  • A. BezdekAffiliated withThe Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary and Department of Mathematics, Auburn University, AL, USA bezdean@mail.auburn.edu
  • , K. BezdekAffiliated withDepartment of Geometry, Eötvös University, Budapest, Hungary kbezdek@ludens.elte.hu
  • , R. ConnellyAffiliated withDepartment of Mathematics, Cornell University, Ithaca, NY 14853, USA {kbezdek,connelly}@math.cornell.edu

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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 D d and A d for d ≥ 3 in E d 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 E d 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.