Browse › Convex and metric geometry › 092 The optimal order of convex-body covering density
Translative covering densities of order n log n
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Abstract
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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Hexagon deposit contributed by Scott Kominers, Multi-coverings by lattice translates of convex bodies (hexagon:2610.00165v1, October 2026). This paper is in its bibliography, but its body text cites only the companion single-lattice paper, whose construction it extends to lattice multi-coverings; the source calls the work "heavily LLM-assisted". [family-level]
Formalization
- OpenAI math release, Lean scope note 092: The optimal order of convex-body covering density (lean/docs/092.md, 2026-10-06). The release's own scope note: the formalization bounds the translative and lattice covering-density suprema, over all convex bodies and over centrally symmetric ones, above and below by absolute multiples of $n\log n$ for large $n$ (comparator file
CoveringDensity.lean). [family-level] - OpenAI, lean/OAI/Geometry/CoveringDensity/Main.lean (openai/math, commit adc7f12). Contains
OAI.CoveringOrder.optimal_order, the two-sided $\Theta(n\log n)$ statement for the four density suprema, together withthetaT_le_thetaL. [family-level]
Related prior literature mentioned by others
- H. Li, X. Liu, Nearly Sharp Bounds for Lattice Coverings by Convex Bodies (arXiv:2607.28429, v2 10 August 2026). The VibeMathed entry for this family describes Theorem 1.1 as extending this paper's order-$n\log n$ lower bound from lattice coverings to translative coverings; the paper cites it as the source of its random-slab and Poisson-witness method.
- Or Ordentlich, Oded Regev and Barak Weiss, New bounds on the density of lattice coverings (arXiv:2006.00340, 2020-05-30). Kalai's October 2026 discussion identifies this earlier work on lattice covering densities as background to the family's two announced results. [family-level]
News and blogs
- Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans (Combinatorics and more, 2026-10-07). Item 7, "Covering density of convex bodies", links both papers of this family and describes this one as giving matching examples "even for unrestricted translative coverings", combined with the universal lattice-covering upper bound. [family-level]
Elsewhere
- Hexagon: Multi-coverings by lattice translates of convex bodies (human and AI; human understanding: all)
- VibeMathed: Problem record
- Emergent Mind: family 092
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Translative-covering-densities-of-order-n-log-n-September-23-2026,
author = {{OpenAI}},
title = {{Translative covering densities of order $n\log n$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf}{OAI:Translative-covering-densities-of-order-n-log-n-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/092.md. The Lean statements have not been checked by people against the paper's claims.
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 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 $c,C\gt 0$.
Other papers in this family:
- A single-lattice covering bound of order n log n · September 23, 2026