Double-lattice packings of convex bodies in the plane
A convex body K in \(E^ 2\) is strictly convex if its boundary contains no line segment. A lattice packing of the plane is a family \(\{K_ i\}\) of convex bodies all congruent to a single convex body K in which each body is a translate of every other and the vectors of the translations form a lattice. A packing \({\mathcal P}\) is called a double-lattice packing if P is the union of two lattice packings \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\) such that the 180\(\circ\) rotation about some point interchanges \({\mathcal P}_ 1\) and \({\mathcal P}_ 2\). A parallelogram inscribed in K is extensive if the length of each of its sides is at least half the length of K in the direction of the side. The authors prove that if K is strictly convex and if \({\mathcal P}\) is a maximum density double-lattice packing with copies of K, then \({\mathcal P}\) is generated by a minimum area extensive parallelogram inscribed in K. They also prove that every convex plane body K admits a double-lattice packing of the plane with density of at least \(\sqrt{3}/2\), extending a result of Mahler and Féjes Tòth.
scientific article; zbMATH DE number 4165900