A Lower Bound on the Density of Sphere Packings via Graph Theory

where the supremum is taken over all packings in R. A celebrated theorem of Minkowski states that ∆n ≥ ζ(n)/2 for all n ≥ 2. Since ζ(n) = 1+ o(1), the asymptotic behavior of the Minkowski bound [9] is given by Ω(2). Asymptotic improvements of the Minkowski bound were obtained by Rogers [10], Davenport and Rogers [4], and Ball [2], all of them being of the form ∆n ≥ cn2, where c > 0 is an absolute constant. The best currently known lower bound on ∆n is due to Ball [2], who showed that there exist lattice packings with density at least 2(n − 1)2ζ(n). In this paper, we use results from graph theory to prove the following theorem.