ABSTRACT
In this chapter we continue our investigation of totally separable packings from a volumetric point of view. First, we outline the recent solution of the contact number problem for smooth strictly convex domains in E 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq779.tif"/> . We discuss this approach in details based on angular measure, Birkhoff orthogonality, Birkhoff measure, (smooth) Birkhoff domains, and approximation by (smooth strictly convex) Auerbach domains, which are topics of independent interests as well. In the next part of this chapter, we connect the study of totally separable packings of discrete geometry to Oler’s inequality of geometry of numbers. More concretely, we discuss an analogue of Oler’s inequality for totally separable translative packings in E 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq780.tif"/> and then use it for finding the highest density of totally separable translative packings (resp., for finding the smallest area convex hull of totally separable packings by n translates) of an arbitrary convex domain in E 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq781.tif"/> . Finally, as a local version of totally separable packings, we introduce the family of ρ-separable translative packings of o-symmetric 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/eq782.tif"/> . In particular, we investigate the fundamental problem of minimizing the mean i-dimensional projection of the convex hull of n non-overlapping translates of an o-symmetric convex body C forming a ρ-separable packing 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/eq783.tif"/> for given d > 1, n > 1, and C.
