The optimal order of convex-body covering density

Determines the optimal worst-case covering density as \Theta(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\gt 0. (Lean)

Manuscripts (2)

A single-lattice covering bound of order n log n

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026}

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.

Translative covering densities of order n log n

September 23, 2026 · source · cite as \cite{OAI:Translative-covering-densities-of-order-n-log-n-September-23-2026}

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.

Result 092 · Convex and metric geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.