ABSTRACT
Summary. In this chapter we present selected proofs of some of the theorems in Chapter 4. In particular, in Section 9.1 we prove estimates for c(n, 3) and c fcc(n). In Section 9.2 we treat the general case, i.e., we examine the contact numbers of translative packings of an arbitrary convex body in E d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq2205.tif"/> . In Sections 9.3 and 9.4 we establish a connection between contact numbers of ball packings and the densities of packings of soft balls, and prove a Blichfeldt-type estimate for the latter quantities, respectively. In Section 9.5 we solve the maximum packing density problem for soft Euclidean disks. In Section 9.6 we prove a Rogers-type bound for soft packings in E 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq2206.tif"/> . Then we turn our attention to totally separable packings. In Section 9.7 we determine the values of c sep(n, 2) and prove bounds for c sep(n, 3). In Section 9.8 we prove estimates for c ℤ(n, d), while in Section 9.9 we examine the general problem of finding c sep(n, d). In Section 9.10 we determine the Hadwiger numbers of strictly convex and smooth convex bodies in E d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq2207.tif"/> for d ≤ 4, and apply these results to prove estimates for c sep(K, n, d) with d ≤ 4. In Section 9.11 we construct topological disks with arbitrarily large Hadwiger numbers, and in Sections 9.12 and 9.13 we prove upper bounds for the Hadwiger numbers of starlike disks in the centrally symmetric and in the general case, respectively.
