Is the worst-case translative covering density of convex bodies linear in the dimension?

For a convex body $K\subset\mathbb R^n$ let $\theta_T(K)$ be the least density of a covering of $\mathbb R^n$ by translates of $K$ . Rogers (1957) proved $\theta_T(K)\le n\log n+n\log\log n+5n$ for every convex body, and lower bounds of order $n$ are known for balls. Naszodi recorded the question whether a universal linear bound holds: is there an absolute constant $C$ with $\theta_T(K)\le Cn$ for every convex body in every dimension? Equivalently, what is the order of the worst-case covering density in dimension $n$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Discrete and convex geometry; translative and lattice coverings
Posed by: M. Naszodi, Flavors of Translative Coverings, New Trends in Intuitive Geometry, Bolyai Soc. Math. Stud. 27 (2018), Section 3 (arXiv 1603.04481)
Year posed: 2016
Years open: 10y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Translative paper, Theorem 1.1: for absolute $c>0$ and all large $n$ there is a centrally symmetric convex body (even a rational polytope) with $\theta_T(K_n)>cn\log n$ , so no universal linear bound exists; with Rogers' bound the worst-case order is $n\log n$ . It extends the lattice lower bound of Li and Liu (2026) to arbitrary translative coverings. The companion proves every convex body has a single-lattice covering of density at most $Cn\log n$ , removing the $(\log\log n)^{O(1)}$ loss of Li-Liu, so the lattice supremum also has order $n\log n$ . Constants are not optimized; Euclidean balls are not treated.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the translative paper (centrally symmetric $K_n$ with $\theta_T(K_n)>cn\log n$ ) was read against the question as the paper states it with the Naszodi reference. Proofs not refereed. The challenge ComparatorChallenges/CoveringDensity.lean is not in lean/formalization.yaml; it is found through lean/docs/092.md and its solution module OAI.Geometry.CoveringDensity.Main exists at the pinned commit. Its statement was read here: absolute $0<c<C$ and $n_0$ such that for $n\ge n_0$ the suprema of translative and of lattice covering density, over all convex bodies and over centrally symmetric ones, lie between $cn\log n$ and $Cn\log n$ , with coverings defined from scratch. This states the headline. Not rebuilt here. SingleLatticeCovering.lean (in the catalogue) states the companion's lattice upper bound.

Is the worst-case translative covering density of convex bodies linear in the dimension?

What was actually shown

What the AI did

Verification

Sources

Discussion