Family 092Convex and metric geometryLeanNot yet digested
The optimal order of convex-body covering density
Determines the optimal worst-case covering density as , for both lattice and unrestricted translative coverings. Every convex body in ℝ, n ≥ 2, admits a lattice covering of density at most ; centrally symmetric examples in every sufficiently large dimension require at least even without the lattice restriction, for absolute .
Manuscripts in this family · 2
A single-lattice covering bound of order n log n
Sep 23, 2026·paper.pdfLean
Every convex body in ℝ, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most , for an absolute constant C. No symmetry or boundary regularity is assumed.
Translative covering densities of order n log n
Sep 23, 2026·paper.pdf
For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds , where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.