math.MG — Metric Geometry

The 24-cell conjecture in the all-contact case and near D4

Contributed by Tejasvi Singh Tomar

The 24-cell conjecture states that every Voronoi cell of a packing of unit balls in $\mathbb{R}^4$ has volume at least $8$, the volume of the regular 24-cell. It implies that no packing of equal balls in $\mathbb{R}^4$ is denser than the lattice packing $D_4$. We prove the conjecture for two classes of cells; in each class, equality holds only for the $D_4$ root configuration. The first consists of the cells whose ball touches every ball centred at distance less than $2\sqrt2$ from its centre, and its proof uses a certificate at the second level of the Lasserre hierarchy. The second consists of the cells whose $24$ nearest centres lie in an explicit neighbourhood of the $D_4$ configuration, whatever the other centres are. For them the volume is at least $8$ plus a fixed multiple of the total outward displacement of the $24$ centres, which we prove by a quantitative rigidity estimate and a second-order expansion of the volume about $D_4$. Our proofs for both classes are computer-assisted and partly formalised in Lean.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23720963