| # The optimal order of convex-body covering density | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [A single-lattice covering bound of order $n\log n$](../../preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf) | |
| - [Translative covering densities of order $n\log n$](../../preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf) | |
| ## Scope | |
| The formalized result gives an absolute constant $C>0$ such that every convex body $K\subset\mathbb R^n$, $n\ge2$, admits a covering by translates along one full-rank lattice $L$ with density $|K|/\mathrm{covol}(L)\le Cn\log n$. The covering is exact, and no symmetry, boundary regularity, or volume normalization is assumed. | |
| The formalization determines the order of the largest covering density in dimension $n$. For all sufficiently large $n$, the suprema of translative covering density and lattice covering density are each bounded above and below by absolute positive multiples of $n\log n$, both over all convex bodies and over centrally symmetric convex bodies. | |
| ## Comparator links | |
| | Result | Comparator statement | | |
| | --- | --- | | |
| | Single-lattice covering bound | [SingleLatticeCovering.lean](../ComparatorChallenges/SingleLatticeCovering.lean) | | |
| | Optimal order for translative and lattice covering-density suprema | [CoveringDensity.lean](../ComparatorChallenges/CoveringDensity.lean) | | |