Browse › Convex and metric geometry › 100 Cylinder coverings below the half-area bound
Finite angular cylinder covers below the half-area bound
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Ricardo Pinho, An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (hexagon:2610.00304v2, v1 17 Oct 2026, v2 18 Oct 2026). Uses a finite tilt in this paper's angular-sector framework, with rational radial cutoffs, to give a 4096-cylinder triangular-base cover of normalized area about 0.499613830783714, certified by exact LP dual witnesses; it cites all four family preprints. The source states that the text is primarily AI-generated and that no optimality is claimed. [AI-generated per source]
- Ricardo Pinho (with contributions by Alexander Richard Towell), tetrahedron-cylinder-cover (GitHub) (code repository, October 2026). Certificates and two independent exact checkers for the 4096-cylinder cover; the README compares it with this paper's small-tilt example (saving about 5.4×10⁻¹⁰ with 16,000,000 cylinders). The README also states that the release's
TriangularCovering.leanencodes the quantities but is not by itself a checked proof of the paper, its final comparator theorem containingsorry. [AI-generated per source]
Formalization
- OpenAI math release, Lean scope notes for family 100: Cylinder coverings below the half-area bound (lean/docs/100.md, 2026). Describes the release's Lean statements for this paper: finite triangular-base covers of the regular tetrahedron with the $1/2-(13/6000)\varepsilon^2+O(\varepsilon^4)$ expansion, and the directionwise counterexample, linking the aggregate and triangular-cover comparator statements; the affine extension to all tetrahedra lies outside the selected statements. [family-level]
- Erdős Problems library, Openai 2026 finite angular cylinder covers below half area bound (verification report, 8 October 2026). Reports rebuilding the release's aggregate and triangular-cover declarations and checking their axioms, distinguishing the aggregate's explicit constants from the qualitative triangular statement and excluding the all-tetrahedra affine extension.
Related prior literature mentioned by others
- Károly Bezdek and Alexander E. Litvak, Packing convex bodies by cylinders (arXiv:1507.05115v2, 21 November 2015; Discrete Comput. Geom. 55 (2016) 725–738). The Erdős Problems library's angular-cover discussion connects this paper's multiple-cover lower bounds to the surviving directionwise one-third bound.
Digests and expositions
- Latent Seminar, Finite cylinder coverings below the tetrahedral half-area bound (6 October 2026). Explains the four manuscripts' constructions, approximation hypotheses and two area normalizations, while expressly disclaiming independent verification of the complete proofs. [family-level]
- heyprof, Cylinders below the half-area bound (3D) (interactive exposition, undated; accessed 21 October 2026). Illustrates the angular construction with adjustable tilt and sector count, distinguishing sampled gap tests at larger tilts from the theorem's small-parameter range.
Elsewhere
- Hexagon: An explicit finite cylinder cover of the regular tetrahedron saving $3.86\times10^{-4}$ below the half-area bound (AI-generated; human understanding: some)
- VibeMathed: Problem record
- Emergent Mind: family 100
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026,
author = {{OpenAI}},
title = {{Finite angular cylinder covers below the half-area bound}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf}{OAI:Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/100.md. The Lean statements have not been checked by people against the paper's claims.
Family 100: Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Other papers in this family:
- Finite cylinder approximation of ruled sets · September 27, 2026
- Slope-field perturbations of the two-cylinder covering · September 27, 2026
- Finite triangular approximation of radial sweeps · September 27, 2026