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Plank theorems and their applications: a survey

William Verreault

TL;DR

The survey collects a broad tapestry of plank problems, starting from Tarski's foundational question on coverings by planks and Bang's subsequent solution, and then moves through Ball's plank theorem for symmetric convex bodies to a spectrum of complex, spherical, and polynomial analogues. It highlights unifying methods—Bang's lemma, linear-algebraic reductions, and the polynomial approach—that yield sharp bounds and enable extensions to infinite-dimensional settings and diverse spaces. The work then links plank theorems to deep applications in analysis (polarization constants, uniform boundedness, dynamics), harmonic analysis (Nazarov-type coefficient problems), and number theory (Diophantine approximation, sphere packings), illustrating the cross-pollination of discrete-geometry ideas with analytic and number-theoretic techniques. Overall, the paper demonstrates that lower bounds for plank coverings not only solve geometric questions but also drive substantive advances across multiple mathematical disciplines, often via translation of geometric constraints into algebraic, analytic, or combinatorial frameworks.

Abstract

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.

Plank theorems and their applications: a survey

TL;DR

The survey collects a broad tapestry of plank problems, starting from Tarski's foundational question on coverings by planks and Bang's subsequent solution, and then moves through Ball's plank theorem for symmetric convex bodies to a spectrum of complex, spherical, and polynomial analogues. It highlights unifying methods—Bang's lemma, linear-algebraic reductions, and the polynomial approach—that yield sharp bounds and enable extensions to infinite-dimensional settings and diverse spaces. The work then links plank theorems to deep applications in analysis (polarization constants, uniform boundedness, dynamics), harmonic analysis (Nazarov-type coefficient problems), and number theory (Diophantine approximation, sphere packings), illustrating the cross-pollination of discrete-geometry ideas with analytic and number-theoretic techniques. Overall, the paper demonstrates that lower bounds for plank coverings not only solve geometric questions but also drive substantive advances across multiple mathematical disciplines, often via translation of geometric constraints into algebraic, analytic, or combinatorial frameworks.

Abstract

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.
Paper Structure (37 sections, 40 theorems, 140 equations, 8 figures)

This paper contains 37 sections, 40 theorems, 140 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Tarski's plank problem
  4. The genesis of plank problems
  5. Partial solution to Tarski's problem
  6. Bang's solution and conjecture
  7. A solution to Tarski's plank problem
  8. A digression on Bognár's trick
  9. Bang's generalization of Tarski's problem
  10. A plank theorem of Ball
  11. Bang's conjecture for centrally symmetric convex bodies
  12. Finite-dimensional Banach spaces
  13. Reduction of the plank theorem to a linear algebra problem
  14. Proof of \ref{['thm Ball V2']}
  15. Infinite-dimensional Banach spaces
  16. ...and 22 more sections

Key Result

Proposition 2.1

The lateral area of a spherical segment formed by the intersection of a sphere of radius $r$ with the region between two parallel hyperplanes separated by distance $d$, both of which intersect the sphere, is $2\pi rd$.

Figures (8)

  • Figure 1: Optimal and nonoptimal positioning of six unit planks on a circular table.
  • Figure 2: Example of the width of a convex body in the direction of a hyperplane in $\mathbb{R}^2$.
  • Figure 3: An example from \ref{['lem1Bang']}. The intersection $(C-u)\cap(C+u)$ contains $\kappa\cdot C$.
  • Figure 4: Labeling of the segments appearing in Hunter's construction.
  • Figure 5: Five zones of equal width covering the unit sphere $S^2$
  • ...and 3 more figures

Theorems & Definitions (59)

  • Proposition 2.1
  • proof : Proof of a special case of Tarski's problem
  • Theorem 2.2
  • Lemma 2.3
  • proof : Proof of \ref{['lem1Bang']}
  • Lemma 2.4
  • Lemma 2.5: Bang's lemma
  • proof : Proof of \ref{['lemBangV2']}
  • proof : Proof of \ref{['Lem2Bang']}
  • Conjecture 2.6: Bang's conjecture
  • ...and 49 more