Finite sphere packing and sphere covering
By a convex body we mean a compact convex subset of Euclidean d-space with non-empty interior. Let k be a given positive integer. A central problem of finite packing and covering is to find (A) the minimum volume of all convex bodies containing a packing of k unit balls, and (B) the maximum volume of all convex bodies that can be covered by k unit balls. L. Fejes Tóth and the third author stated the conjectures that, for \(d\geq 5\), the following two configurations give the solutions to the problems (A) and (B) respectively: The centres of the k balls are equidistantly spaced on a straight line such that (A) the convex hull of the balls has minimum volume or, (B) the balls cover a convex body of maximum volume. In the present paper several partial results are given which support both conjectures. Furthermore, some relations between finite and infinite packings and coverings are considered. For this purpose, the authors introduce a special concept of density suited for packings and coverings with respect to the whole space as well as for finite packings and coverings. Some upper and lower bounds for the volume of a parallel body of a regular simplex are developed which are of independent interest. \{Reviewer's remark: On p. 39 there are three mistakes. G(s) has to be replaced twice by \(V(sQ+B^ d)-V(S_{G(s)}+B^ d)\) (1.7 from below), and \(G(16)=23969\) (1.5 from below).\}
scientific article; zbMATH DE number 4075951
scientific article