Notions of denseness

Abstract

The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent and weakly recurrent dense packings, and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, such a packing is limit-equivalent to a completely saturated one. Diffusive dominance is yet sharper than complete saturation and leads to a better understanding of n-saturation.

Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.html

References in corpus (1)

• An overview of the Kepler conjecture

Cited by in corpus (13)

• New upper bounds on sphere packings I
• New upper bounds on sphere packings II
• Counterintuitive ground states in soft-core models
• Density bounds for outer parallel domains of unit ball packings
• The local optimality of the double lattice packing
• Periodicity and Circle Packing in the Hyperbolic Plane
• The Honeycomb Conjecture
• Uniqueness and Symmetry in Problems of Optimally Dense Packings
• Sphere Packings in Hyperbolic Space: Periodicity and Continuity
• On the existence of completely saturated packings and completely reduced covering
• On densest packings of equal balls of $\rb^{n}$ and Marcinkiewicz spaces
• Optimally dense packings of hyperbolic space
• Mesh ratios for best-packing and limits of minimal energy configurations