Dr. Martin Henk Combinatorial Optimization Finite and Innnite Packings at the Konrad-zuse-zentrum in Berlin. I Want to Thank Günter M. Ziegler and the Institute for Their Logistical and Technical Support. I Am Grateful to Finite and Infinite Packings

Preface In this thesis we give a new approach to the classical problems of finite and infinite packings and lattice packings of convex bodies. This approach is based on the introduction of parameterized densities δ ρ (K, C), ρ ∈ R >0. The parameterized density of a finite packing set C of a convex body K with respect to the parameter ρ is defined by δ ρ (K, C) = #(C)V (K) V (conv(C) + ρK) , where V (·) denotes the volume and #(C) denotes the cardinality of the packing set C. In particular for ρ = 1 we obtain a measure for the quality of finite packings, that is equivalent to the " densities " used by Rogers, Groemer, Oler and L. Fejes Tóth. For a given convex body K, ρ ∈ R >0 , and an integer n we are asking for a packing set C of cardinality n, that has a maximal parameterized density δ ρ (K, C). Moreover we are interested in the asymptotic behavior of these maximal parameterized densities δ ρ (K, n) with respect to n, i.e. we study δ ρ (K) = lim sup n→∞ δ ρ (K, n). The " density " δ ρ (K) may be regarded as a " limit density " of finite packings of K with respect to the parameter ρ. For large values of ρ it turns out that the limit density δ ρ (K) is equal to δ(K), the density of a densest infinite packing. So the parameterized densities build a bridge between the finite and infinite packing problem. The least upper bound of the set of parameters ρ with δ ρ (K) = δ(K) plays a special role in our investigation and is called critical parameter ρ c (K). In order to characterize δ ρ (K) for small parameters ρ, we transfer the notion of " sausage " packing sets, introduced by L. Fejes Tóth for finite packings of balls, to arbitrary convex bodies. A packing set S n (K) is called a densest sausage configuration of K with cardinality n, if the volume of the convex hull of S n (K)+ K is minimal among all packing sets C of cardinality n and dim(C) = 1. The limit δ s ρ (K) = lim n→∞ δ ρ (K, S n (K)) may be considered as the parameterized density of a densest " infinite …