This chapter discusses finite translative arrangements of a given convex domain in ℝ. For packings, we determine the translative packing density and the asymptotic structure of the optimal packings of n translates for large n. We prove the analogous results for coverings by translates of centrally symmetric convex domain K, and, moreover, the hexagon bound for coverings of any convex shape by at least seven translates of K. The main tools are the Oler inequality for packings and the Fejes Tóth inequality for coverings. In addition, we prove the Bambah inequality for coverings when the convex hull of the centres is covered. Many of these proofs are based on constructing Delone type simplicial complexes. The final part of Chapter 2 discusses problems related to the Hadwiger number of K, which is the maximal number of nonoverlapping translates of K touching K.

Assuming that K is centrally symmetric, let us compare our knowledge of arrangements of translates of K (see this chapter) and of congruent copies of K (see Chapter 1). We have essentially the same results for packings. However, we know much more about coverings by translates because any two homothetic convex domains are noncrossing (see Section 1.5).

About the Minkowski Plane

Convex domains K and C are called homothetic if C = x + λK for some λ > 0. This chapter discusses problems when the geometry of the plane is considered in terms of convex domains homothetic to a given one.