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

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometrymath.OC — Optimization and Controlmath.NT — Number Theory

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

The 24-cell conjecture states that every Voronoi cell of a packing of unit balls in R4\mathbb{R}^4 has volume at least 88, the volume of the regular 24-cell. It implies that no packing of equal balls in R4\mathbb{R}^4 is denser than the lattice packing D4D_4. We prove the conjecture for two classes of cells; in each class, equality holds only for the D4D_4 root configuration. The first consists of the cells whose ball touches every ball centred at distance less than 222\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 2424 nearest centres lie in an explicit neighbourhood of the D4D_4 configuration, whatever the other centres are. For them the volume is at least 88 plus a fixed multiple of the total outward displacement of the 2424 centres, which we prove by a quantitative rigidity estimate and a second-order expansion of the volume about D4D_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

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Version history

  1. v1 · 24 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23720963CurrentNov 12, 2026