Definition
Tarski's plank problem is a conjecture in convex geometry concerning the covering of a convex body in Euclidean space by a collection of planks. A plank is defined as the region of space bounded by two parallel hyperplanes; its width is the distance between these hyperplanes.
The problem asks: given a convex body $K \subset \mathbb{R}^n$ and a finite family of planks ${P_i}_{i=1}^m$ that together cover $K$, is it necessarily true that
$$ \sum_{i=1}^{m} w(P_i) ;\ge; w(K), $$
where $w(P_i)$ denotes the width of plank $P_i$ and $w(K)$ denotes the minimal width of $K$ (the smallest distance between two parallel supporting hyperplanes of $K$)?
Historical background
- The problem was posed by the Polish logician and mathematician Alfred Tarski in the 1930s, originally as a question in geometric measure theory.
- In 1951, the Hungarian mathematician Tibor Gallai (often cited as Bang) proved the conjecture in full generality; the result is now known as Bang's theorem or Bang’s plank theorem.
- The theorem has since inspired numerous extensions and related results, notably in the work of Keith Ball (1991) on the reverse Brunn–Minkowski inequality and the Ball–Bang theorem concerning symmetric convex bodies.
Statement of Bang’s theorem
Let $K$ be a convex body in $\mathbb{R}^n$ and let ${P_i}_{i=1}^m$ be a collection of planks that cover $K$. Then
$$ \sum_{i=1}^{m} w(P_i) \ge w(K). $$
The inequality is sharp; equality holds, for example, when $K$ is a line segment and the planks are arranged collinearly to exactly fill its length.
Key concepts and terminology
- Convex body: A compact convex set with non‑empty interior in Euclidean space.
- Width of a convex body: The infimum of the distances between parallel supporting hyperplanes of the body; equivalently, the minimal distance across the body in any direction.
- Supporting hyperplane: A hyperplane that touches a convex set but does not intersect its interior.
Proof outline (Bang, 1951)
Bang’s original proof proceeds by induction on the dimension $n$. The central idea is to select a direction in which the convex body attains its minimal width and to analyze the projection of the covering planks onto the orthogonal line. By comparing the total projected lengths of the planks with the width of the projection of $K$, Bang establishes the required inequality. The induction step reduces the problem to a lower‑dimensional case by intersecting the configuration with a suitable hyperplane.
Generalizations and related results
- Ball’s theorem (1991) – For centrally symmetric convex bodies, the same inequality holds, and the bound can be strengthened in certain symmetric settings.
- Complex and spherical versions – Analogous plank covering problems have been formulated for spherical geometry and for complex vector spaces; partial results are known.
- Fractional and probabilistic versions – Recent work investigates random plank coverings and fractional covering numbers, extending the deterministic inequality to expected values.
- Higher‑codimension coverings – Variants where coverings are made by neighborhoods of affine subspaces of codimension greater than one have been studied, though the direct analogue of Tarski’s inequality does not always hold.
Applications
- Geometric tomography: Estimating body dimensions from limited directional data.
- Optimization: Bounds for covering problems in computational geometry.
- Functional analysis: Connections with norms and Banach space geometry via Ball’s extensions.
References
- T. Bang, “A solution of Tarski’s plank problem,” Proceedings of the American Mathematical Society, vol. 2, no. 4, 1951, pp. 990–993.
- K. Ball, “The reverse isoperimetric problem for convex bodies and a characterization of the Euclidean ball,” Inventiones Mathematicae, vol. 104, 1991, pp. 401–418.
- R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014.
See also
- Covering problems in discrete geometry
- Width (convex geometry)
- Bang’s theorem
- Ball–Bang theorem