The local optimality of the double lattice packing (Q312160)

A packing in \(\mathbb{R}^n\) by copies of a convex body \(K\) is a family \(\{\varphi(K): \varphi \in H\}\) of non-overlapping images of \(K\) under suitable Euclidean isometries \(\varphi \in H\). It is a double lattice packing if \(H\) is a group generated by \(n\) translations with linearly independent translation vectors and one point reflection. \textit{G. Kuperberg} and \textit{W. Kuperberg} [Discrete Comput. Geom. 5, No. 4, 389--397 (1990; Zbl 0709.52007)] and \textit{D. M. Mount} [in: Discrete and computational geometry: Papers from the DIMACS special year. Proceedings of workshops, held at DIMACS from September 1989 through August 1990. Providence, RI: American Mathematical Society; New York: Association for Computing Machinery. 245--262 (1991; Zbl 0744.68069)] characterize densest double lattice packings of a convex polygon \(P\) in the plane. The paper under review gives a criterion for an arbitrary packing \(\{\varphi(K): \varphi \in H\}\) in \(\mathbb{R}^n\) to be a local maximizer of density within the space of all packings generated by \(K\). Under mild additional assumptions, the above mentioned densest double lattice packings of a convex polygon \(P\) turn out to be strict local maximizers among all packings generated by \(P\). In particular, this applies to the densest double lattice packings by regular pentagons and heptagons.