Sphere packing

A collection of points \(\mathbf{P}\) in \(\mathbb{R}^d\) such that any two pairwise distinct points are at least a unit distance apart. Such an arrangement often has the notion of a point density. This is given by

\[\begin{align} \limsup_{R \to \infty } \frac{\mathbf{P}\cap B_{R}(0)}{\mathop{\mathrm{vol}}(B_{R}(0))} \end{align}\]

If we arrange balls of radius 1/2 centered at each one of these points, we can also talk about the packing efficiency. This is just \(\mathop{\mathrm{vol}}(B_{1/2}(0))\) times the point density.

The sphere packing problem, is to find the largest possible value of this packing efficiency, in \(d\)-dimensions, and the point distributions that maximize this.

In case where the point-set \(\mathcal{P}\) is a lattice, we call it a lattice packing