Tarski's Plank Problem¶

Relate geometric coverage to directional thickness: any finite family of Euclidean planks covering a convex body must have total plank width at least the body's minimum width.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2933
Origin domain: mathematics
Subdomain: convex geometry
Aliases: Plank problem, Tarski plank theorem, Bang's plank theorem

Core Idea¶

Tarski's plank problem asks whether covering a convex body in Euclidean space by planks requires at least as much total plank width as the body's minimum width. A plank is the closed region between two parallel hyperplanes, and its width is the perpendicular distance between those hyperplanes. A convex body's width in a direction is the distance between its two supporting hyperplanes orthogonal to that direction; its minimum width is the infimum over directions. If planks P_1,...,P_m cover a convex body K, the theorem states sum_i w(P_i) >= w(K).

Alfred Tarski posed the question, and Thøger Bang proved the affirmative result. Bang's 1951 paper A Solution of the Plank Problem is the primary source.^[1] The title problem is historical: the base Euclidean conjecture is solved and should not be described as open. Modern work studies stronger, affine, relative-width, centrally symmetric, complex, or restricted-covering variants. Bezdek's review situates Bang's theorem and later plank problems while distinguishing the original width inequality from its extensions.^[2]

The theorem's force lies in comparing unlike directions through one additive lower bound. Each plank may be oriented independently. A cover can overlap heavily, extend infinitely beyond the body, or use many narrow pieces. Nevertheless, the sum of their perpendicular widths cannot drop below the thinnest directional width of the convex body. For a Euclidean ball of diameter d, minimum width is d; no finite cover by planks of widths totaling less than d succeeds, regardless of their orientations.

The support-function formulation clarifies normalization. For a unit vector u, the width of K in direction u is h_K(u)+h_K(-u), where h_K(u)=sup{x dot u : x in K}. A plank perpendicular to u has width equal to the difference of its bounding support levels. Unit normalization is essential: replacing u by a scaled vector changes support values unless the denominator is adjusted. The theorem assumes ordinary Euclidean distance and a convex body, conventionally compact, convex, and with nonempty interior.

The candidate survives collision review against generic covering, feasible regions, partitions, Kakeya sets, and the affine plank problem. It is not simply cover a set: it binds one special covering primitive, independently oriented hyperplane strips, an additive resource, Euclidean width, and a sharp lower bound tied to the target's minimum width. Coverage / Reachability is the strict prime parent because every point of the convex body must lie in at least one plank; Tarski's theorem adds a quantitative cost certificate for that complete reach.

Structural Signature¶

The Euclidean ambient space. A finite-dimensional real Euclidean space fixes perpendicular distance and hyperplanes.
The convex body. A compact convex set with nonempty interior is the target of the cover.
The directional width. Parallel supporting hyperplanes measure the body's thickness in each unit direction.
The minimum width. The infimum of directional widths supplies the target lower-bound quantity.
The plank primitive. Each covering set lies between two parallel hyperplanes and has their perpendicular distance as width.
The finite cover. The union of the planks contains every point of the convex body.
The independent orientations. Different planks may point in different directions.
The additive covering cost. Individual plank widths are summed without subtracting overlaps.
The Bang inequality. Total plank width is at least the body's minimum width.
The sharpness frame. A single plank aligned with a minimum-width direction can attain equality for the base bound.

What It Is Not¶

Not an open conjecture in its base form. Bang proved Tarski's Euclidean statement.
Not area or volume covering. The cost is the sum of one-dimensional widths, not covered measure.
Not a partition. Planks may overlap and extend beyond the body.
Not a parallel-only cover. Plank orientations can differ.
Not the affine plank problem. Relative-width and affine-invariant strengthenings require separate statements.
Not the Kakeya problem. Kakeya sets contain segments in directions rather than cover a convex body by planks.
Not a statement for arbitrary nonconvex targets without modification. Convexity and minimum width are constitutive.

Scope of Application¶

Tarski's plank problem is literal when a convex body is covered by finite Euclidean planks and the sum of their widths is compared with the body's minimum width.

Convex covering theory. Establishing lower bounds on strip-like covering resources.
Extremal geometry. Studying equality and near-equality configurations.
Support functions. Recasting width and planks through linear functionals.
Centrally symmetric bodies. Comparing stronger or affine variants under symmetry.
Normed spaces. Formulating analogues only after redefining width and dual normalization.
Discrete geometry education. Demonstrating that arbitrary orientations do not evade an additive width bound.
Relative-width problems. Measuring plank width against the body's width in the same direction in later conjectures.
Separation arguments. Using convexity and supporting hyperplanes to certify unavoidable covering cost.

Clarity¶

A clear statement defines convex body, supporting hyperplane, directional width, minimum width, plank, plank width, finite cover, and ambient metric before giving the inequality. Use unit normals or divide support differences by the normal's norm. State whether boundaries are included; for the standard finite result this convention does not change width but affects literal containment language. Distinguish the historical conjecture from Bang's theorem and cite the result as solved. If relative width w(P)/w(K,u) appears, label the affine plank problem or another extension. Do not infer an area bound from the theorem. In worked cases, calculate the body's minimum width rather than substituting a convenient diameter that may not be minimal.

Manages Complexity¶

A covering by differently oriented infinite strips appears resistant to a one-direction projection: a plank narrow in its own normal direction may look broad in another. The theorem compresses that multidirectional geometry into an additive invariant. Convexity supplies supporting hyperplanes and prevents thin disconnected appendages from gaming the target width. The lower bound ignores overlaps, locations, and order, retaining only the plank widths and target minimum. That simplicity is exactly why extensions are delicate. Relative widths, non-Euclidean norms, complex planks, or coverings of nonconvex sets alter the invariant. A reference-grade proof or application records which geometry makes additivity meaningful rather than carrying the inequality across variants by analogy.

Abstract Reasoning¶

Specify the Euclidean space and convex target with its compactness and interior assumptions.
Represent directional thickness through supporting hyperplanes or a normalized support function.
Minimize over unit directions to obtain the target's minimum width.
Represent every covering set as a plank with a declared normal and two boundary levels.
Compute each plank's perpendicular width under the same Euclidean normalization.
Verify that the union covers the entire convex body, not only its boundary or sampled points.
Sum widths without overlap correction because the theorem prices covering resources, not covered measure.
Apply Bang's inequality to compare total width with the target minimum.
Check equality or slack and identify whether a minimum-direction plank realizes the bound.
If assumptions change, restate the relevant plank variant instead of importing the base theorem.

Knowledge Transfer¶

The theorem transfers a powerful pattern: define a directional resource cost for each covering primitive, identify a target invariant, and prove that complete coverage forces additive cost at least that invariant even when primitives are differently oriented. Similar reasoning informs geometric inequalities and duality, but the numerical statement does not transfer without the Euclidean width structure. The abstraction teaches that overlap and directional freedom do not necessarily reduce a conserved covering budget.

Examples¶

Canonical¶

Let K be a disk of diameter ten. Every directional width of the disk is ten, so its minimum width is ten. Suppose finitely many planks of arbitrary orientations cover the disk. Bang's theorem gives sum_i w(P_i) >= 10. Ten unit-width planks may meet the bound only if arranged to cover; nine unit-width planks cannot cover, regardless of overlap or angle. One plank of width ten already covers when aligned across the disk, showing the lower bound is sharp.^[1]

Mapped back: disk of minimum width ten + arbitrary plank cover → sum of perpendicular plank widths → Bang lower bound ten → sharp single-plank witness.

Applied / In Practice¶

For an elongated convex body, the analyst first computes widths from its support function and identifies the thinnest direction. A proposed cover uses several diagonal planks because each seems to span much of the body. Plotting area coverage is irrelevant to the theorem. After normalizing each plank's hyperplane equation, their perpendicular widths sum to less than the body's minimum width, so the proposal is impossible without checking placement point by point. Bezdek's review shows how later plank questions modify this comparison rather than invalidate it.^[2]

Mapped back: support-function widths → minimum target thickness → normalized diagonal plank widths → additive impossibility certificate.

Structural Tensions¶

Directional freedom vs. invariant lower bound. Planks rotate independently. Diagnostic: Are all widths measured perpendicular to their own boundaries?
Coverage vs. measure. Infinite planks can have unbounded volume. Diagnostic: Is cost being measured by width rather than area or volume?
Overlap freedom vs. additive cost. Planks may overlap without discount. Diagnostic: Has any overlap correction been incorrectly subtracted?
Historical problem vs. present theorem. The name retains a question form. Diagnostic: Is the base Euclidean statement being called unresolved?
Euclidean statement vs. affine extensions. Relative-width variants look similar. Diagnostic: Does the formula use absolute or body-relative width?
Convenient width vs. minimum width. A chosen direction may be thicker. Diagnostic: Was the infimum over directions actually taken?
Convex target vs. arbitrary set. Convexity supports the theorem. Diagnostic: Is the target a convex body under the stated definition?

Structural–Framed Character¶

The structure is a convex target, directional and minimum width, plank primitives, finite union coverage, additive widths, and a sharp inequality. The frame is dimension, Euclidean metric, target geometry, plank orientation, and chosen variant. The theorem is insensitive to placement details once coverage is certified, but highly sensitive to changing the width definition or convexity assumptions.

Structural Core vs. Domain Accent¶

The transferable core is complete target coverage by resource-priced primitives → additive cost lower-bounded by a target invariant. The domain accent is Euclidean hyperplanes, convex bodies, support width, planks, independent directions, and Bang's minimum-width inequality. Remove that accent and Coverage / Reachability remains; preserve it and Tarski's Plank Problem is a distinct convex-geometric theorem package.

Coverage / Reachability is the strict parent by specialization. The plank family must reach every point of the convex body, and any uncovered point defeats the premise. Tarski's theorem adds Euclidean plank geometry and a quantitative width lower bound; Coverage / Reachability is broader and carries no such metric cost.

The prospective workspace queue contains one strict upward edge to prime:coverage_reachability. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Tarski's Plank Problem Domain-specific

Parents (1) — more general patterns this builds on

Tarski's Plank Problem is a kind of Coverage / Reachability Prime

Coverage / Reachability is the strict parent by specialization.

Hierarchy paths (2) — routes to 2 parentless roots

Tarski's Plank Problem → Coverage / Reachability → Completeness

Show alternative path (1)

Neighborhood in Abstraction Space¶

Tarski's Plank Problem sits in a sparse region of the domain-specific corpus (76^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Metric Geometry & Approximation (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Affine Plank Problem. Stronger relative-width or affine-invariant formulation.
Kakeya Set. Contains a unit segment in every direction rather than covering a body with planks.
Strip Covering of a Plane. May target the entire plane or periodic sets under different resource measures.
Partition. Disjoint exhaustive decomposition rather than overlapping cover.
Lebesgue Covering Dimension. Topological invariant unrelated to summed plank width.
Width of a Poset. Order-theoretic antichain size, not geometric thickness.
Plank in Woodworking. Physical inspiration for the name, not the mathematical primitive.

References¶

[1] Thøger Bang, A Solution of the ‘Plank Problem’, Proceedings of the American Mathematical Society 2, no. 6 (1951): 990–993, https://doi.org/10.1090/S0002-9939-1951-0046672-4. registry ↩a ↩b

[2] Károly Bezdek, Tarski's Plank Problem Revisited, in Geometry—Intuitive, Discrete, and Convex (Springer, 2013), https://doi.org/10.1007/978-3-642-41498-5_2. registry ↩a ↩b