Remark: Family 092. 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 ${\mathbb{R}}^n$, $n\ge 2$, 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 $c,C>0$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.