Extremum properties of lattice packing and covering with circles
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Peter M. Gruber
Abstract
Recent results on extremum properties of the density of lattice packings of smooth convex bodies and balls extend and refine Voronoĭ’s classical criterion for balls. This article treats in more detail the special case of lattice packings and coverings with circular discs. The aim is to determine those lattices for which the densities of the corresponding packings and coverings with circular discs, and certain products and quotients thereof, are semi-stationary, stationary, extreme, and ultra-extreme. The latter notion is a sharper version of extremality. It turns out that in all cases where solutions exist, the regular hexagonal lattices are solutions. Unexpectedly, in a few cases the square lattices and in one case special parallelogram lattices are solutions too. A further surprise is the fact that the lattices forwhich the circle packing density is extreme coincide with the lattices with ultra-extreme density. For semi-stationarity, stationarity and ultra-extremality the duality between packing and covering results breaks down. All results may be interpreted in terms of binary positive definite quadratic forms.
© 2016 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Pseudospherical surfaces of low differentiability
- Complex Finsler structures on tensor products
- On the Lefschetz trace formula for Lubin–Tate spaces
- Classes of generalized Weingarten surfaces in the Euclidean 3-space
- On Sasaki–Ricci solitons and their deformations
- How many torsionless affine connections exist in general dimension?
- A counterexample to the containment I(3) ⊂ I2 over the reals
- On the polyhedrality of global Okounkov bodies
- Extremum properties of lattice packing and covering with circles
- Simple crystallizations of 4-manifolds
- Sharply 2-transitive groups
Articles in the same Issue
- Frontmatter
- Pseudospherical surfaces of low differentiability
- Complex Finsler structures on tensor products
- On the Lefschetz trace formula for Lubin–Tate spaces
- Classes of generalized Weingarten surfaces in the Euclidean 3-space
- On Sasaki–Ricci solitons and their deformations
- How many torsionless affine connections exist in general dimension?
- A counterexample to the containment I(3) ⊂ I2 over the reals
- On the polyhedrality of global Okounkov bodies
- Extremum properties of lattice packing and covering with circles
- Simple crystallizations of 4-manifolds
- Sharply 2-transitive groups
