This chapter covers quite a number of topics because finite translative arrangements have been rather intensively investigated. Actually, one more topic is parametric density, which is the subject of Chapter 10.

Given a convex body K in ℝd, Section 9.1 introduces the associated Minkowski space, namely, the normed space induced by K0 = (K – K)/2. The main body of the chapter starts with density-type problems. We characterize K when some translates of K tile a certain convex body (see Theorem 9.2.1). Then, appealing to methods in Chapter 7, we investigate the asymptotic behavior of optimal packings and coverings of a large number of translates of K in Sections 9.3 and 9.4. We also describe the classical economic periodic packings and coverings by translates of K (see Theorem 9.5.2).

The next topic is the so-called Hadwiger number H(K): the maximal number of nonoverlapping translates of K that touch K. Theorem 9.6.1 says that λd < H(K) ≤ 3d – 1, where λ > 1 is an absolute constant. The lower bound H(K) ≥ d2 + d is verified as well (see Theorem 9.7.1), which is optimal if d = 2, 3. For positive α, Sections 9.8 to 9.10 discuss a natural generalization of the Hadwiger number, that is, the maximal number Hα(K) of nonoverlapping translates of αK touching K. We note that H∞(K) is related to antipodal sets and equilateral sets (see Section 9.11).