On Tightest Packings in the Minkowski Plane

is satisfied for any two distinct points P, Q of L. Since the work of Thue on circle packings in the plane, many people have noted that the mesh of a lattice is the inverse of the density of its points and have asked whether there are arbitrary pointsets of maximal density satisfying the packing condition. C. A. Rogers [8] showed in 1951, for example, that for the Euclidean plane no packing is denser than the critical lattice. The question which first was taken up by Thue concerns itself with the metric characteristics of finite pointsets of R satisfying the Euclidean distance packing condition. In this paper we develop the motivation for giving the metric characterization in terms of invariant measures and we solve the Thue problem (which was solved for plane packings in Jordan polygons by N. Oler [3] in 1961) for the vertex set of arbitrary finite plane simplicial complexes meeting the rf-packing condition (cf. Theorem 4), thereby extending the method used by Folkman and Graham [2] in the case of Euclidean distance. It is interesting to observe that the plane packings of a given finite number of points of minimal slackness measure (cf. [9]) must be part of a critical 2-lattice if the unit ball defining d is strictly convex. On the other hand there are infinite irregular packings of maximal density which are not subsets of a critical 2-lattice. Of course, the corresponding questions can be considered in more than two dimensions. As a consequence of the work of Minkowski, it is clear what the invariant measures for convex bodies must be,