On lattice packings of mirror or centrally symmetric convex three-dimensional body

V. V. Makeev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 29-38

Résumé

Proved is a number of statements concerning lattice packings of mirror 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 is as follows. Every three-dimensional mirror symmetric convex body admits a lattice packing with density $\ge8/27$. Moreover, 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): every three-dimensional centrally symmetric convex body admits a lattice packing with density $>0.53835$. In this paper, it is only proved that every three-dimensional centrally symmetric convex body admits a lattice packing with density $(\sqrt{3}+ \sqrt[4]{3/4} + 1/2 )/6 > 0.527$.