Local Covering Optimality of Lattices : Leech Lattice versus Root Lattice E 8

The Leech lattice is the exceptional lattice in dimension 24. Soon after its discovery by Leech [9], it was conjectured that it is extremal for several geometric problems in R: the kissing number problem, the sphere packing problem, and the sphere covering problem. In 1979, Odlyzko and Sloane and independently Levenshtein solved the kissing number problem in dimension 24 by showing that the Leech lattice gives an optimal solution. Two years later, Bannai and Sloane showed that it gives the unique solution up to isometries (see [6, Chapters 13, 14]). Unlike the kissing number problem, the other two problems are still open. Recently, Cohn and Kumar [5] showed that the Leech lattice gives the unique densest lattice sphere packing in R up to scaling and isometries. Furthermore, they showed that the density of any sphere packing (without restriction to lattices) in R cannot exceed the one given by the Leech lattice by a factor of more than 1 + 1.65 · 10. At the moment it is not clear how one can prove a corresponding result for the sphere covering problem. In this paper we take a first step into this direction by showing the following theorem.