OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
Determines the optimal worst-case covering density as Θ(nlog n), for both lattice and unrestricted translative coverings. Every convex body in ℝ^(n), n ≥ 2, admits a lattice covering of density at most Cnlog n; centrally symmetric examples in every sufficiently large dimension require at least cnlog n even without the lattice restriction, for absolute c,C> 0.
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Every convex body in ℝ^(n), n ≥ 2, admits a covering by translates along one full-rank lattice with density at most Cnlog n, for an absolute constant C. No symmetry or boundary regularity is assumed.
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For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds c nlog n, where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.
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