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 $c,C\gt 0$.
released 2026-09-23 | 1 theorem · 8 lemmas · 10 proofs · 8,671 words |
PLAY LEVEL 1 »(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.
released 2026-09-23 | 1 theorem · 6 lemmas · 17 proofs · 11,073 words |
PLAY LEVEL 2 »(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.