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 [9] who proved that for n = 3 the packing density of the regular octahedron is at most 1 − 1.4 . . .× 10. In this paper, we prove upper bounds for the packing density of the n-dimensional regular cross-polytope in the case that n ≥ 7. We use a modification of Blichfeldt’s method [2] due to G. Fejes Tóth and W. Kuperberg [7].