Result 092, Convex and metric geometry

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 $c,C\gt 0$ .

Lean formalization New or sharp bound

The bigger picture

Why it matters

Covering space with shifted copies of one solid shape inevitably creates overlap. These manuscripts claim that the worst-case overlap needed grows like dimension times its logarithm, even when the copies must follow a single repeating grid spanning all directions.

What changes?

The manuscripts report that every convex body, a compact solid containing the segment between any two points, in dimension n at least two admits a lattice covering of density at most C times n log n. Density measures average overlap. No symmetry or boundary regularity is assumed. In every sufficiently large dimension, centrally symmetric examples require density exceeding c times n log n even with unrestricted translation locations. The positive constants c and C are absolute.

What does that help mathematicians do?

Together, these bounds would fix the optimal worst-case growth rate for both lattice and unrestricted translative coverings. They rule out any universal bound proportional to dimension alone, even for bodies symmetric about a center. Researchers could also deduce that allowing arbitrary placement instead of a single lattice cannot improve the worst-case order, although it may help for particular bodies.

Are there practical applications?

The immediate value is foundational: the claims identify how efficiently copies of an arbitrary convex body can cover high-dimensional space, and how much overlap some shapes force. They provide matching benchmarks for covering theory. The abstracts do not describe a practical placement algorithm or demonstrate an application outside mathematics.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A single-lattice covering bound of order n log n

September 23, 2026 23 pages Main result formalized in Lean

Every convex body in ℝⁿ, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most $Cn\log n$ , for an absolute constant C. No symmetry or boundary regularity is assumed.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026,
  author = {{OpenAI}},
  title = {{A single-lattice covering bound of order $n\log n$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf}{OAI:A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026}},
  year = {2026}
}

Translative covering densities of order n log n

September 23, 2026 30 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/092.md.

The optimal order of convex-body covering density

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

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

At a glance

Subject: Convex and metric geometry
Manuscripts: 2
Pages: 53
Dated: Sep 23, 2026
Lean formalization: Yes
Reasoning summary: No

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