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

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.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Is the worst-case translative covering density of convex bodies linear in the dimension? · Linear covering density question

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

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.