Rogers' bound

Remarks

C. A. Rogers proved the bound in 1958. The proof splits space into the Voronoi cells of the centres, and splits each cell further into simplices. A local volume argument shows that the balls fill at most the fraction $\sigma_n$ of each Voronoi cell. For $n = 2$, the triangle of side $2$ has area $\sqrt 3$, and the three discs cover three sectors of angle $\pi/3$, of total area $\pi/2$. So $\sigma_2 = \pi/\sqrt{12}$, and the bound is sharp by Thue's theorem. For $n = 3$ one gets $\sigma_3 \approx 0.7797$, a little above the true optimum $\pi/\sqrt{18} \approx 0.7405$ of the Kepler conjecture.

Going deeper

For a student

Rogers' bound was the best general upper bound on sphere packing density for about twenty years. For large $n$ it is roughly $\tfrac{n}{e} 2^{-n/2}$, and the Kabatiansky–Levenshtein bound improved it exponentially in 1978 by a method based on linear programming. In dimensions $4$ to $36$ the Cohn–Elkies linear programming bound is better, and it is sharp in dimensions $8$ and $24$.