Worst-case unrestricted translative covering density
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How large can unrestricted translative covering density be for a convex body in dimension n? Translates need not form one lattice. Rogers’ general upper bound has order n log n.
Why it matters and what counts as progress
Why it matters: A matching lower bound would determine the worst-case order and exclude a universal linear bound.
What counts as progress: Convex bodies with unrestricted density at least c n log n in every sufficiently large dimension, or an improved universal upper bound; lattice-only examples do not suffice.
How the problem developed
- 23 Sept 2026 · Reported lower bound [1]
OpenAI Reported centrally symmetric convex bodies in every sufficiently large dimension whose unrestricted translative covering density is at least c n log n, matching the order of the classical upper bound.
AI contribution
Translative covering densities of order n log n
For every sufficiently large dimension n, an OpenAI manuscript constructs a centrally symmetric convex body whose unrestricted translative covering density exceeds c n log n for an absolute c>0. With Rogers’ O(n log n) upper bound, this fixes the worst-case unrestricted order and rules out a universal O(n) bound.
Sources
- OpenAI, Translative covering densities of order n log n
- Translative covering densities of order n log n
- OpenAI, translative covering lower-bound theorem and context
- OpenAI, translative covering lower-bound conclusion
- OpenAI, translative covering lower-bound manuscript
- OpenAI, Lean theorem for translative coverings
- OpenAI, covering-density comparison statement
- OpenAI Math, release and provenance account
Website timeline
Publication dates and editorial changes.
OpenAI reports a superlinear translative covering-density lower bound