The optimal order of convex-body covering density

What OpenAI claims

Determines the optimal worst-case covering density as Θ(n log n), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most Cn log n; centrally symmetric examples in every sufficiently large dimension require at least cn log n even without the lattice restriction, for absolute c,C > 0. (Lean)

Papers (2)

1. A single-lattice covering bound of order n log n

   23 September 2026 · Preprint, not peer reviewed · PDF

   Every convex body in ℝn, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most Cn log n, for an absolute constant C. No symmetry or boundary regularity is assumed.

2. Translative covering densities of order n log n

   23 September 2026 · Preprint, not peer reviewed · PDF

   For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds c n log n, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.