Optimal lattice packing density of the four-dimensional cross-polytope

Prove that the optimal lattice packing density of the four-dimensional cross-polytope C₄ is δ*(C₄) = 32/45, and characterize all lattices attaining this density as the lattice generated by the four explicitly specified vectors, up to coordinate permutations and coordinate sign changes.

Background

The paper proves that the four-dimensional cross-polytope has lattice kissing number κ*(C₄) = 40 and exhibits a lattice attaining this number with packing density 2/3. It also exhibits another lattice, with density 32/45, in which every cross-polytope touches 26 others.

The authors conjecture that 32/45 is the optimal lattice packing density and that the displayed lattice is, modulo coordinate permutations and sign changes, the unique optimizer. Thus the density problem remains unresolved despite the exact determination of the lattice kissing number.

References

Conjecture 4.1. In $\mathbb{E}4$, we have $$\delta*(C_4)=32/45,$$ and it can be attained if and only if the corresponding lattice is generated by ${(-\frac{3}{4},\frac{3}{4},\frac{1}{4},\frac{1}{4}),$ $(\frac{1}{4},-\frac{3}{4},\frac{3}{4},\frac{1}{4}), (\frac{1}{4},\frac{1}{4},-\frac{3}{4},\frac{3}{4}), (\frac{3}{4},\frac{1}{4},\frac{1}{4},-\frac{3}{4})},$ up to permutation of coordinates and signs of coordinates.

— On Generalized Kissing Numbers of Convex Bodies (II)  (2501.06792 - Li et al., 12 Jan 2025) in Section 4, immediately after the proof of Theorem 3