Dense ellipsoid packings (Q1598805)

Let \(\delta_{d}^{L}\), \(\delta_{d}^{C}\) denote the density of the densest lattice packing of spheres or ellipsoids, and the supremum of packing densities of congruent ellipsoids in \(\mathbb E^{d}\), respectively. In [Ser. Discrete Math. Theor. Comput. Sci. 4, 71-80 (1991; Zbl 0738.52019)] \textit{A. Bezdek} and \textit{W. Kuperberg} proved that \(\delta_{d}^{C}>\delta_{d}^{L}\), for all \(d\geq 3\), and obtained the lower bound \(\delta_{3}^{C}\geq 0.7533\dots\) in dimension three. The author presents constructions of dense ellipsoid packings in \(\mathbb{E}^{d}\), for \(4\leq d\leq 8\), and provides lower bounds for \(\delta_{d}^{C}\) in these dimensions. The idea of the method is to fill the tunnels left by a lattice or non-lattice packing with suitable translates of an ellipsoid. The lower bounds for \(\delta_{d}^{C}\) are then obtained from the known values of \(\delta_{d}^{L}\). It turns out that in \(\mathbb{E}^{8}\) there are packings with congruent ellipsoids which are more than \(42.9\%\) denser than the densest lattice packing.