Bang's half-area cylinder covering question (and the 1-codimensional cylinder covering conjecture)

A cylinder in $\mathbb R^3$ is $B+\mathbb Ru$ with $u$ a unit vector and $B\subset u^\perp$ of finite area. Bang's plank theorem says plank widths in a covering of a convex body sum to at least its minimal width; for cylinders, the two-cylinder cover of a regular tetrahedron has total base area exactly half the minimum projection area $A_{\min}(K)$ . Bezdek and Litvak, attributing the example to Bang (1951), asked whether every finite cylinder covering of a convex body $K\subset\mathbb R^3$ satisfies $\sum_i|B_i|\ge\frac12A_{\min}(K)$ ; Bezdek and Khan conjectured the stronger directionwise bound $\sum_i|B_i|/|\pi_{u_i^\perp}K|\ge\frac12$ . Do these half-area lower bounds hold?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Convex geometry, covering problems
Posed by: Thoger Bang (example); Karoly Bezdek and Alexander Litvak; Karoly Bezdek and Muhammad Khan (directionwise form)
Year posed: 1951
Years open: 75y
Solved: 2026-09-27
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 18 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: a regular tetrahedron $K$ with $A_{\min}(K)=\sqrt2$ has, for each $0<\varepsilon\le1/2000$ , a cover by $2\lceil2/\varepsilon^2\rceil$ cylinders with compact triangular perpendicular bases and $\frac1{\sqrt2}\sum|B_i|=\frac12-\frac{13}{6000}\varepsilon^2+O(\varepsilon^4)$ , strictly below half. Corollary 1.2: every nondegenerate tetrahedron has a finite cover with directionwise relative cost below 1/2, refuting the Bezdek-Khan conjecture. Not shown: the optimal constant (Bezdek-Litvak's lower bound 1/3 stands) or counterexamples among centrally symmetric bodies.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts, all dated September 27, 2026: the principal angular-sector construction and three companions giving alternative or more general constructions (ruled-set approximation, slope-field perturbations, radial sweeps).

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction, history section and Theorem 1.1 with Corollary 1.2 were read against the question as Bezdek and Bezdek-Khan state it. Lean: the challenge TriangularCovering (theorem OAI.TriangularCovering.main, solution module OAI.Geometry.TriangularCovering.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. It states, for the regular tetrahedron of edge 2, $A_{\min}=\sqrt2$ , finite triangular-base cylinder covers with normalized area $1/2-(13/6000)\varepsilon^2+O(\varepsilon^4)$ and total area below $A_{\min}/2$ , and a cover with directionwise relative cost below 1/2. That is the headline counterexample. The affine extension to all tetrahedra is outside it. Not rebuilt here.

Bang's half-area cylinder covering question (and the 1-codimensional cylinder covering conjecture)

What was actually shown

What the AI did

Verification

Sources

Discussion