Densest packings of typical convex sets are not lattice-like
The packing density \(\delta(C)\) of a set \(C\) with finite positive Lebesgue measure is defined as the supremum of the densities of all packings of congruent copies of \(C\). The lattice-packing density \(\delta_L (C)\) of \(C\) is the supremum of the densities of all lattice packings of \(C\). The author proves: If \(P\) is a convex polygon which has no parallel sides, then \(\delta_L(P) < \delta (P)\). There is a pleasing discussion of related results (e.g., generalized Dirichlet cells, covering) and open problems.
scientific article; zbMATH DE number 778436