Local covering optimality of lattices: Leech lattice versus root lattice $E_8$

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice $E_8$. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of $E_8$. The new lattice yields a sphere covering which is more than 12\% less dense than the formerly best known given by the lattice ${A_8}^*$. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a nonsimplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid, our answer is even the strongest possible in a sense.