We start this chapter by discussing the Newton number: the maximal number of nonoverlapping unit balls in ℝd touching a given unit ball. The three-dimensional case was the subject of the famous debate between Isaac Newton and David Gregory, and it was probably the first finite packing problem in history.

In the later part of the chapter, optimality of a finite packing of n unit balls means that the volume or some mean projection of the convex hull is minimal. If the dimension d is reasonably large then the packing minimizing the volume of the convex hull is the sausage; namely, the centres are collinear. However, if some mean projection is considered then the convex hull of the balls in an optimal arrangement is essentially some ball for large n in any dimension. For the mean width, we also verify that, in the optimal packing of d + 1 balls, the centres are vertices of a regular simplex.

Concerning optimal coverings of compact convex sets by n unit balls in, mostly conjectures are known; namely, it is conjectured that the optimal coverings are sausage-like (see Section 8.6). However, sound density estimates will be provided when a larger ball is covered by unit balls.

In this chapter only a few proofs are provided because the arguments either use the linear programming bound (6.1) or are presented in Chapter 7 for packings and coverings by congruent copies of a given convex body.