Lattices of optimal finite lattice packings (Q1769065)

The densest finite lattice packings of \(n\) balls of parametric density \(\rho>\rho_C(B^d)\), where \(\rho_C(B^d)\) is a positive constant, are shown to be subsets of critical lattices for all large enough cardinalities \(n\). The result follows for all \(n\) in two dimensions and is conjectured to be true for all \(n\) in all dimensions. On the other hand, there exist certain centrally symmetric convex octagons, say \(K\), such that their lattices are not critical for \(K\) for some \(\rho>\rho_C(K)\). The constants \(\rho_C(B^d)\) and \(\rho_C(K)\) are defined in the paper.