ABSTRACT
The well-known “kissing number problem” asks for the largest number of non-overlapping unit balls touching a given unit ball in the Euclidean d-space 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/eq575.tif"/> . Generalizing the kissing number, the Hadwiger number or the translative kissing number H(K) of a convex body K 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/eq576.tif"/> is the maximum number of non-overlapping translates of K that all touch K. In this chapter we study and survey the following natural extension of these problems. A finite translative packing of the convex body K 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/eq577.tif"/> is a finite family P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq578.tif"/> of non-overlapping translates of K 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/eq579.tif"/> . Furthermore, the contact graph G( P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq580.tif"/> ) of P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq581.tif"/> is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. The number of edges of G( P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq582.tif"/> ) is called the contact number of P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274572/b797b7b9-7287-418a-b2e8-56a9266a9de2/content/eq583.tif"/> . Finally, the “contact number problem” asks for the largest contact number, that is, for the maximum number c(K, n, d) of edges that a contact graph of n non-overlapping translates of K can have 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/eq584.tif"/> . In the first half of this chapter, we survey the bounds proved for c(K, n, d) using volumetric methods. Then we turn our attention to an important subfamily of translative packings called totally separable packings. Here a packing of translates of a convex body K 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/eq585.tif"/> is called totally separable if any two packing elements can be separated by a hyperplane of 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/eq586.tif"/> disjoint from the interior of every packing element. In the second half of this chapter, we study the analogues of the Hadwiger and contact numbers for totally separable translative packings of K labelled by H sep(K) and c sep(K, n, d) and survey the bounds proved for H sep(K) as well as c sep (K, n, d) using volumetric ideas. This chapter is a revised and strongly extended version of [41].
