Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a hundred-year-old lower bound on the maximal packing density due to Minkowski in d-dimensional Euclidean space ℝ d . The asymptotic behavior of this bound is controlled by 2-d in high dimensions. Using an optimization procedure that we introduced earlier [Torquato and Stillinger 02] and a conjecture concerning the existence of disordered sphere packings in ℝd, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by 2-0.77865... d , thus providing the putative exponential improvement of Minkowski's bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in ℝ d for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonne...
['Salvatore Torquato', 'Frank H. Stillinger']
New Conjectural Lower Bounds on the Optimal Density of Sphere Packings