The stealthiest particle configurations in 8 and 24 dimensions

We show that the $E_8$ and Leech lattices generate the stealthiest isometry-invariant, locally square-integrable point processes of intensity $1$ in ${\mathbb{R}}^8$ and ${\mathbb{R}}^{24}$, respectively, and that they are the unique stealthiest processes. We also apply the proof technique to sphere packing and energy minimization, and we construct a $20$-dimensional periodic configuration that is stealthier than any known $20$-dimensional lattice with the same particle density. This periodic configuration is a formal dual of Vardy’s $20$-dimensional sphere packing.