Lattice Packings of Mirror Symmetric or Centrally Symmetric Three-Dimensional Convex Bodies

Abstract

We prove a number of statements concerning lattice packings of mirror symmetric or centrally symmetric convex bodies. This enables one to establish the existence of sufficiently dense lattice packings of any three-dimensional convex body of such type. The main result states that each three-dimensional, mirror symmetric, convex body admits a lattice packing with density at least 8/27. Furthermore, two basis vectors of the lattice generating the packing can be chosen parallel to the plane of symmetry of the body. The best result for centrally symmetric bodies was obtained by Edwin Smith (2005): Each three-dimensional, centrally symmetric, convex body admits a lattice packing with density greater than 0.53835. In the present paper, it is only proved that each such body admits a lattice packing with density $$ \left(\sqrt{3}+\sqrt[4]{3/4}+1/2\right)/6>0.527 $$ . Bibliography: 5 titles.