On Packing of Minkowski Balls

Abstract

We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and  Chebyshev–Cohn balls. We investigate lattice packings of  these balls on  planes with varying Minkowski metric and  search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding  balls.