Extremum properties of lattice packing and covering with circles

Abstract

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.