Sausages are good packings
In a previous paper [J. Reine Angew. Math. 453, 165-191 (1994; Zbl 0797.52010)] the authors introduced a new original approach to the study of finite and infinite packings and were able to prove several important results (e.g. they confirmed Fejes Tóth's ``sausage'' conjecture in high dimensions). Their approach is primarily based on the following parametric density: Let \(C_n\) be a set of \(n\) points defining a packing arrangement of the unit ball \(B^d\) in \(d\)-dimensional Euclidean space; then, for \(\rho > 0\) the density \(\delta (B^d, C_n, \rho)\) of the packing is defined to be the ratio of \(n\) times the volume of \(B^d\) over the volume of the body \(\text{conv } C_n + \rho B^d\). In the present paper, the authors prove that, for \(\rho < \sqrt {2}\) and sufficiently high dimensions, ``sausage'' packings of balls (that is packings for which \(C_n\) is contained in a line segment) minimize the density \(\delta (B^d, C_n, \rho)\). This gives an improvement of the bound \(\rho < \sqrt {3}\) proved in the quoted article. The above density can easily be generalized to arbitrary convex bodies, and in this case the bound \(\sqrt {2}\) has to be replaced by a constant depending on the dimension of the space and on the inradius and circumradius of the given body. Moreover, it is shown how results for infinite packings can be deduced by this density.
scientific article; zbMATH DE number 764228
scientific article