Locally dense finite lattice packings of spheres

We consider finite packings of unit-balls in Euclidean 3-spaceE³ where the centres of the balls are the lattice points of a lattice polyhedronP of a given latticeL³E³. In particular we show that the facets ofP induced by densest sublattices ofL³ are not too close to the next parallel layers of centres of balls. We further show that the Dirichlet-Voronoi-cells are comparatively small in this direction. The paper was stimulated by the fact that real crystals in general grow slowly in the directions normal to these dense facets.The results support, to some extent, the hypothesis that real crystals grow preferably such that they need little volume, i.e that they are locally dense.Dedicated to A. Florian on the occasion of this 60th birthday