Covering a tetrahedron with less cylinder area than expected

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Family 100 · Convex and metric geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Imagine covering a solid tetrahedron with infinitely long tubes. Each tube may point in its own direction, and its cross-section need not be circular. How small can the sum of the cross-sectional areas be?

The answer cannot be compared directly with the tetrahedron’s volume: area and volume have different units. A natural reference is its smallest shadow under orthogonal projection onto a plane. A familiar cover of a regular tetrahedron uses two tubes parallel to opposite edges, with total cross-sectional area exactly half that smallest shadow.

Was one half a universal lower bound? Family 100 reports that it is not. By allowing many slightly different directions, the manuscripts claim finite covers of a regular tetrahedron with cost strictly below the proposed threshold. The improvement is small, but its sign settles the proposed inequality. Two companion approximation theorems explain how continuously varying directions can be replaced by finitely many genuine cylinders without losing the saving.

2. Background and essential definitions

A convex body in $\mathbb R^3$ is a compact convex set with nonempty interior. A cylinder here means

$$C=B+\mathbb R u,$$

where $u$ is a unit vector and $B$ is a measurable planar set in the perpendicular plane $u^\perp$. Its cost is the area $|B|$. The whole infinite cylinder is used; no height enters its cost. In particular, a triangular or parallelogram base is allowed.

Write $\pi_{u^\perp}K$ for the orthogonal shadow of $K$ in that plane, and define

$$A_{\min}(K)=\min_{|u|=1}|\pi_{u^\perp}K|.$$

For a finite cover $K\subseteq\bigcup_i(B_i+\mathbb R u_i)$, the half-area question proposed

$$\sum_i|B_i|\geq\frac12 A_{\min}(K).$$

Overlap is counted repeatedly in the sum. This is a covering problem, so overlap is allowed and gaps are forbidden.

A second formulation measures each tube against the shadow in its own direction:

$$R(\mathcal C;K)=\sum_i\frac{|B_i|}{|\pi_{u_i^\perp}K|}.$$

The proposed bound $R\geq1/2$ is stronger, since

$$R(\mathcal C;K)\leq\frac{\sum_i|B_i|}{A_{\min}(K)}.$$

Thus a cover below the absolute half-area threshold violates both statements. The directionwise ratio is invariant under invertible affine transformations: a linear change of coordinates multiplies a base and the body’s matching shadow by the same planar area factor. The minimum-shadow normalization does not have this general affine invariance.

3. Development history and previous partial results

The ancestor is Tarski’s plank problem. A plank is the region between two parallel planes, and its cost is their separation. Bang’s plank theorem says that the total width of planks covering a convex body is at least its minimum width. Ball later established a stronger direction-dependent plank inequality for centrally symmetric bodies.

Cylinders ask for the analogous comparison one dimension further down: a planar base replaces a one-dimensional width. The geometry changes because different axes can cooperate in ways unavailable to parallel-sided planks.

Bezdek’s Tarski’s plank problem revisited, Problem 3.1, attributes both the half-area question and the two-cylinder tetrahedral example to Bang. Bezdek and Litvak subsequently proved a directionwise lower bound of $1/3$ for arbitrary three-dimensional convex bodies, and the stronger bound $1$ for ellipsoids. These results show that the total relative cost cannot approach zero, while leaving a gap between the general theorem and the conjectured $1/2$.

Bezdek and Khan formulated the directionwise version within their codimensional cylinder covering conjectures. The regular tetrahedron seemed especially informative: its elementary two-cylinder cover attained the proposed constant. But exhibiting equality in a simple construction does not show that the construction is optimal when the number and directions of cylinders are unrestricted.

The new manuscripts focus precisely on that remaining freedom. Their proposed counterexamples keep the tetrahedron and change the cover, exploiting a small area gain from tilting while carefully preserving overlap.

4. What the new manuscripts claim

Finite counterexamples with ordinary polygonal bases

M1 gives explicit finite covers with compact triangular bases. In one convenient normalization, the tetrahedron is

$$K=\{(x,y,\sqrt2\,t):0\leq t\leq1,\ |x|\leq1-t,\ |y|\leq t\}.$$

Its four vertices form a regular tetrahedron of edge length $2$, and $A_{\min}(K)=\sqrt2$. For $0<\varepsilon\leq1/2000$, M1 states a construction with $2\lceil2/\varepsilon^2\rceil$ cylinders and normalized cost

$$\frac{\sum_i|B_i|}{\sqrt2} =\frac12-\frac{13}{6000}\varepsilon^2+E(\varepsilon), \qquad |E(\varepsilon)|\leq2\varepsilon^4.$$

The negative quadratic term wins over the bounded fourth-order error in the stated range. The claim concerns a finite cover for each chosen parameter, not merely a limiting continuum of tubes.

M3 supplies alternative constructions using slope fields. It states that every regular tetrahedron admits finite covers below both half thresholds with compact parallelogram bases, or alternatively bounded open measurable bases. Affine invariance then gives $R<1/2$ for every nondegenerate tetrahedron. This extension is specifically directionwise; it does not assert the absolute minimum-shadow inequality for every affine image.

Turning a slope field into finitely many cylinders

M2 studies a compact label set $D\subset\mathbb R^2$, a continuously differentiable field $V$ near $D$, and the swept set

$$E(D,V,L)=\{(s,p+sV(p)):p\in D,\ |s|\leq L\},\qquad L>0.$$

The derivative matrix $DV$ describes how neighboring slopes vary. Under the pointwise conditions

$$\operatorname{tr}DV=0,\qquad -L^{-2}<\det DV\leq0,$$

M2 claims finite cylinder covers with total perpendicular area arbitrarily close from above to

$$\int_D\frac{dp}{\sqrt{1+|V(p)|^2}}.$$

The intercept tiles are nondegenerate squares; their perpendicular projections are parallelograms. The theorem allows arbitrary compact $D$, including positive-area boundaries, and does not require the derivative’s operator norm to be small. In particular, a large shear with $(DV)^2=0$ can satisfy its conditions. This is an approximation theorem; producing a counterexample also requires choosing fields whose sweeps cover the tetrahedron at a low enough integral cost.

A radial alternative with triangular bases

M4 gives a separate approximation theorem for rays in an intercept plane. Choose independent planar vectors $a,b$, a unit normal $e$, a continuous radial cap $R(t)\geq0$, and a continuously differentiable planar direction correction $w(t)$. The sweep consists of

$$p_0+\rho(a+tb)+s(e+w(t)), \quad t\in[\alpha,\beta],\quad 0\leq\rho\leq R(t),\quad |s|\leq L.$$

Its key alignment condition is $w'(t)=\lambda(t)(a+tb)$, with continuous real $\lambda$. For every partition of the parameter interval, and any chosen tag in each subinterval, M4 claims one triangular-base cylinder per subinterval, with the tagged direction. As the largest subinterval length tends to zero, their total cost tends to

$$|\det(a,b)|\int_\alpha^\beta \frac{R(t)^2}{2\sqrt{1+|w(t)|^2}}\,dt.$$

Convergence is uniform over the partitions and tags. The radial cap may vanish. Applied to two suitably overlapping sweeps, this gives another all-triangular finite cover below the half-area threshold. Its prescribed directions and cylinder count are stronger structural features than merely cutting parallelogram bases into triangles.

5. Why the advance matters

Provisional importance: 4/5. The family claims a counterexample to a long-standing covering conjecture and introduces finite approximation methods with uses beyond the particular tetrahedron. Its scope is more specialized than an entire theory of optimal convex-body coverings, but the conceptual change is substantial.

The regular tetrahedron’s most obvious equality construction would no longer identify the correct constant. A successful sharp inequality must account for coordinated variation of cylinder directions, rather than extrapolate from a few symmetric tubes.

There is also a useful lesson about continuous optimization. An integral may suggest a cheaper cover without actually describing a legal finite arrangement. M2 and M4 address that gap explicitly, retaining coverage while controlling the added area. The finite realization is part of the mathematical claim, not a cosmetic reformulation.

6. The main idea

Tilt saves area, but can open gaps

Start with the two-cylinder cover and split its intercept regions into small patches. Assign slightly different slopes to neighboring patches. A patch tilted relative to its cylinder’s perpendicular plane has smaller projected area: the factor is $(1+|V|^2)^{-1/2}$. For small slopes, the saving is quadratic.

The danger is that these tilted tubes no longer fit together. Extra padding restores coverage, but padding consumes the same area that the tilt was meant to save. The constructions arrange the geometry so that the saving exceeds this repair cost.

M1 works with narrow angular sectors. The two complementary families move their separating boundaries in opposite directions at first order. After that cancellation, a controlled second-order comparison leaves a strict gain. Fine enough sectors make discretization errors smaller still.

Fields designed around the boundary

M3 constructs a smooth rational field from a stream function that is constant on wedge walls. A stream function encodes a planar field with zero divergence; its constant boundary values also make the field tangent to those walls. These properties control both the derivative and the coverage geometry.

The complementary families’ inverse labels shift oppositely at first order. Slight enlargements then leave a second-order overlap surplus, while costing less than the projected-area saving. Further constructions use singular power fields and an angular limiting field; their behavior near the apex requires separate treatment.

These are straight segments with a direction frozen at each label, not trajectories following a differential equation. That distinction matters because a flow would generally trace curves rather than cylinder axes.

Why radial alignment helps discretization

In M4, changing the parameter changes the direction initially along the existing intercept ray. This displacement can be absorbed by moving along that ray. What remains transverse to it is only quadratic in the subinterval length.

A slightly enlarged triangle therefore contains all intercepts belonging to one subinterval. Each enlargement costs a small amount, and their total tends to zero as the partition becomes finer. M2 achieves a broader finite approximation through local affine models and coverings by square intercept tiles. The two tools have different hypotheses and geometric guarantees; neither should be read as permitting arbitrary direction fields at no cost.

7. Scope, limitations, and open questions

The main counterexample concerns infinite cylinders in three dimensions with freely chosen directions and measurable bases. It is not a statement about circular cylinders, bounded tube lengths, disjoint packings, or a prescribed small number of cylinders.

The manuscripts do not identify the optimal relative covering constant. Combining their stated tetrahedral counterexample with the earlier general theorem places the infimum for any nondegenerate tetrahedron in $[1/3,1/2)$. Its exact value remains a separate question.

The approximation results also retain real restrictions: M2 has derivative conditions and a strict segment-length bound; M4 requires radial alignment. Their success does not justify replacing an arbitrary continuous family by finite tubes with the same integral cost.

Finally, the displayed improvement is a strict perturbative saving. It does not claim a large numerical reduction or the most economical practical cover. The importance lies in crossing the conjectured boundary and in the mechanisms that make that crossing possible.

8. Sources and further reading

For the historical formulation, see Károly Bezdek, Tarski’s plank problem revisited, especially Problem 3.1. Bezdek and Alexander Litvak, Covering convex bodies by cylinders and lattice points by flats, Journal of Geometric Analysis 19 (2009), 233–243, gives the earlier cylinder inequalities. Bezdek and Muhammad Khan, The geometry of homothetic covering and illumination, records the directionwise conjecture. These distinguish the original area question from its affine-invariant strengthening.

Source manuscripts

• M1. OpenAI, Finite angular cylinder covers below the half-area bound (September 27, 2026).
• M2. OpenAI, Finite cylinder approximation of ruled sets (September 27, 2026).
• M3. OpenAI, Slope-field perturbations of the two-cylinder covering (September 27, 2026).
• M4. OpenAI, Finite triangular approximation of radial sweeps (September 27, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.