Kugellagerungen und Konvexgeometrie. (Sphere packings and convex geometry). (German) Zbl 0688.52004

The author gives an interesting survey on connections between sphere packings/coverings and the classical geometry of numbers. Besides well- known applications of both areas (e.g. in the fields of discrete optimization, crystallography etc.), the interaction of discrete and convex geometry within them deserves a large interest. From this viewpoint, on the base of an extensive list of references the following subjects are briefly discussed: finite and infinite packings/coverings and their connections to lattice point problems, especially finite sphere packings/coverings with the famous sausage phenomena (eventually yielding explanations for the growth of crystals, because sausage phenomena not only occur in the case of spheres), lattice polytopes, estimates for lattice point numbers by means of Minkowski’s quermass-integrals and linear functionals (diameter, thickness). Finally, the author underlines some interesting relations to current research areas, as computational geometry and theoretical information science.
Reviewer: H.Martini


52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H31 Lattice packing and covering (number-theoretic aspects)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
05B40 Combinatorial aspects of packing and covering
11H06 Lattices and convex bodies (number-theoretic aspects)
90C10 Integer programming