ABSTRACT
In this chapter we present selected proofs of some of the theorems in Chapter 5. Continuing our investigation of the contact numbers of totally separable translative packings, in Section 10.1 we determine c sep(D, n, 2) for any Birkhoff domain D. In Section 10.2 we prove that Birkhoff domains are dense in the family of o-symmetric, smooth strictly convex domains, and use this result in Section 10.3 to determine c sep(K, n, 2) for any o-symmetric, smooth strictly convex region. In Section 10.4 we prove an Oler-type inequality for totally separable translative packings. Based on this result, we determine the value of δ sep(K) for any plane convex body K in Section 10.5, and prove sharp bounds for this quantity in the family of plane convex bodies in Section 10.6. We apply these results to find the minimum area of the convex hull of a totally separable packing of n translates of a plane convex body in Section 10.7, showing that the minimum is attained for example for linear packings. In Section 10.8 we prove that for ρ-separable packings, which can be regarded as local versions of totally separable packings, the same does not hold for the mean i-dimensional projection for any i < d if n is suffficiently large, since in this case the minimal mean projection is attained if the convex hull is close to a Euclidean ball.
