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.
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).
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.
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.
Abstract Reasoning¶
- Specify the Euclidean space and convex target with its compactness and interior assumptions. 2. Represent directional thickness through supporting hyperplanes or a normalized support function. 3. Minimize over unit directions to obtain the target's minimum width. 4. Represent every covering set as a plank with a declared normal and two boundary levels. 5. Compute each plank's perpendicular width under the same Euclidean normalization. 6.
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.
Relationships to Other Abstractions¶
Current abstraction Tarski's Plank Problem Domain-specific
Parents (1) — more general patterns this builds on
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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
- Tarski's Plank Problem → Coverage / Reachability → Surjectivity → Function (Mapping)
Neighborhood in Abstraction Space¶
Tarski's Plank Problem sits in a sparse region of the domain-specific corpus (76th 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
- Sphere packing — 0.85
- Reach (Mathematics) — 0.83
- Metric projection — 0.83
- Affine plank problem — 0.82
- Kakeya Set — 0.82
Computed from structural-signature embeddings · 2026-09-08