Result 100, Convex and metric geometry

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How much total cross-sectional area is needed for straight tubes to cover a solid? The unreviewed manuscripts report that a regular tetrahedron can be covered using less than a conjectured geometric minimum.

What changes?

The claimed three-dimensional construction covers the entire closed regular tetrahedron with finitely many cylinders, straight tubes with fixed cross-sections. Their perpendicular bases are compact triangles with total area strictly less than half the tetrahedron's smallest orthogonal projection area, its smallest shadow under parallel light. This contradicts Bang's half-area bound. Affine invariance also gives counterexamples for every nondegenerate tetrahedron to the stronger directionwise half-bound, which normalizes each base area by the corresponding projection area.

What does that help mathematicians do?

The strict inequality rules out using half the minimum projection area as a universal lower bound for finite cylinder covers of three-dimensional convex bodies. Crucially, the construction covers boundary points too and uses only finitely many cylinders, so the failure cannot be attributed to an infinite limiting cover or an omitted boundary. This narrows what a replacement covering inequality could assert.

Are there practical applications?

Its immediate value is foundational: it changes which geometric constraints researchers can impose when comparing a solid's shadows with the cost of covering it by cylinders. The accompanying radial-sweep result supplies a specific mathematical tool: finite triangular-base cylinder covers of radially aligned segment sweeps, with total base area converging to a weighted parameter area as the partition is refined.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

Finite angular cylinder covers below the half-area bound

September 27, 2026 20 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

Finite cylinder approximation of ruled sets

September 27, 2026 12 pages

We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C¹, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Finite-cylinder-approximation-of-ruled-sets-September-27-2026,
  author = {{OpenAI}},
  title = {{Finite cylinder approximation of ruled sets}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf}{OAI:Finite-cylinder-approximation-of-ruled-sets-September-27-2026}},
  year = {2026}
}

Slope-field perturbations of the two-cylinder covering

September 27, 2026 23 pages

The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026,
  author = {{OpenAI}},
  title = {{Slope-field perturbations of the two-cylinder covering}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf}{OAI:Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026}},
  year = {2026}
}

Finite triangular approximation of radial sweeps

September 27, 2026 28 pages

We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Finite-triangular-approximation-of-radial-sweeps-September-27-2026,
  author = {{OpenAI}},
  title = {{Finite triangular approximation of radial sweeps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf}{OAI:Finite-triangular-approximation-of-radial-sweeps-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/100.md.

Cylinder coverings below the half-area bound

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The half-area cylinder-covering conjecture predicts that a cylinder cover of a convex body has total perpendicular-base area at least half its smallest projection area. The formalization constructs finite covers of a regular tetrahedron by cylinders with compact triangular bases whose total area is strictly below that bound. For the explicit small parameter $\varepsilon$ , the normalized area is $1/2-(13/6000)\varepsilon^2+O(\varepsilon^4)$ , giving the strict saving.

It also gives a counterexample to the directionwise normalized half-bound. The broader Comparator file includes the companion ruled-set and radial-sweep approximation results.

The formalization approximates compact ruled families of line segments by finite cylinder covers with perpendicular-base area at most the integral projection cost plus any prescribed positive error. The selected results cover the stated $C^1$ hyperbolic differential condition, square-zero and nilpotent cases, physical rescalings, logarithmic and product-cell estimates, and singular cases. Compact label sets need not have regular boundary.

The differential and segment-length hypotheses remain part of each result. The aggregate Comparator file also contains the companion tetrahedron counterexamples and radial-sweep estimates.

The formalization proves that every regular tetrahedron has a finite cylinder cover whose total perpendicular-base area is strictly less than half its minimum projection area. It supplies covers with parallelogram bases and also proves the directionwise normalized cost is below one half. These are counterexamples to the two cylinder-covering bounds studied in the paper.

The linked aggregate includes the accompanying explicit angular covers and finite approximation theorems for ruled sets and radial sweeps. The affine extension of the directionwise conclusion to every nondegenerate tetrahedron is outside the selected regular-tetrahedron statements.

The formalization proves finite triangular-cylinder approximation for radially aligned segment sweeps. For every tagged partition of the parameter interval, it constructs one cylinder per interval with the prescribed radial direction, and the total perpendicular-base area converges to the weighted parameter area as the mesh tends to zero. It also covers the corresponding actual swept set under the stated geometric hypotheses.

The same aggregate includes finite cylinder covers of regular tetrahedra below half the minimum projection area. These are the radial approximation and covering consequences relevant to the paper.

Comparator links

Result	Comparator statement
Cylinder-covering counterexamples and companion approximations	CylinderCovering.lean
Explicit triangular-cylinder cover below half area	TriangularCovering.lean
Ruled-set approximation within the cylinder-covering aggregate	CylinderCovering.lean
Finite cylinder approximation for the selected ruled families	RuledCovering.lean
Cylinder-covering counterexamples and approximation results	CylinderCovering.lean
Finite triangular approximation of radial sweeps	CylinderCovering.lean