The packing density of the $n$-dimensional cross-polytope

The packing density of the regular cross-polytope in Euclidean $n$-space is unknown except in dimensions $2$ and $4$ where it is 1. The only non-trivial upper bound is due to Gravel, Elser, and Kallus (2011) who proved that for $n=3$ the packing density of the regular octahedron is at most $1-1.4\ldots\times 10^{-12}$. In this paper, we prove upper bounds for the packing density of the $n$-dimensional regular cross-polytope in the case that $n\geq 7$. We use a modification of Blichfeldt's method due to G. Fejes T\'oth and W. Kuperberg (1993).