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 $n\log n$
- Translative covering densities of order $n\log n$
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 |
| Optimal order for translative and lattice covering-density suprema | CoveringDensity.lean |