Draft:Affine plank problem

A plank in ${\displaystyle \mathbb {R} ^{d}}$ is the closed region between two parallel hyperplanes. Given a convex body ${\displaystyle K\subset \mathbb {R} ^{d}}$ (a compact convex set with nonempty interior) and a plank ${\displaystyle P}$ with unit normal vector ${\displaystyle u\in S^{d-1}}$, the relative width of ${\displaystyle P}$ with respect to ${\displaystyle K}$ is

${\displaystyle \operatorname {rw} _{K}(P)={\frac {w(P)}{w_{K}(u)}},}$

where ${\displaystyle w(P)}$ is the width of the plank (the distance between its bounding hyperplanes) and ${\displaystyle w_{K}(u)}$ is the width of ${\displaystyle K}$ in the direction ${\displaystyle u}$ (the distance between the two supporting hyperplanes of ${\displaystyle K}$ perpendicular to ${\displaystyle u}$).

Since ${\displaystyle w_{K}(u)\geq w(K)}$ for every direction ${\displaystyle u}$, the relative width of any plank is at most its absolute width divided by the minimal width of ${\displaystyle K}$. Therefore the affine plank conjecture implies Bang's theorem (the solution to Tarski's plank problem), making it a strictly stronger statement.^[2]

A key feature of relative width is its affine invariance: if ${\displaystyle T}$ is an invertible affine transformation, then the relative width of ${\displaystyle T(P)}$ with respect to ${\displaystyle T(K)}$ equals the relative width of ${\displaystyle P}$ with respect to ${\displaystyle K}$. This invariance is the reason for the name "affine plank problem".^[5]

Alexander (1968) showed that the affine plank conjecture is equivalent to a geometric generalisation of the pigeonhole principle:^[6]

Ball (1991) described this as a "multi-dimensional pigeon-hole principle".^[1] The equivalence connects the affine plank problem to Diophantine approximation via observations of Harold Davenport (1962).^[6][7]

• Tarski's plank problem, the predecessor problem proved by Bang (absolute widths)
• The annulus plank problem, asking whether removing a small homothetic copy from the interior changes the minimal total width needed for covering
• The Pyjama problem, on covering the plane by rotated periodic stripes
• The Fejes Tóth zone conjecture, proved by Jiang and Polyanskii in 2017,^[11] which states that zones covering the unit sphere have total width at least ${\displaystyle \pi }$; its proof uses ideas from Bang's lemma
• Conway's fried potato problem, solved by András Bezdek and Károly Bezdek (1995),^[12] which is related to the equivalent "pigeonhole" formulation

Through its connections to Bang's lemma and its equivalent formulations, the affine plank problem and its partial solutions have applications in several areas:

• Diophantine approximation: The equivalent pigeonhole formulation yields results on simultaneous approximation of real numbers, sharpening estimates of Davenport.^[1][7]
• Sphere packings: Ball used his plank theorem to give a lower bound for the density of lattice packings of balls.^[2]
• Polynomial zero avoidance: Glazyrin, Karasev, and Polyanskii (2023) developed a polynomial approach that unifies Tarski's plank problem with its complex and spherical variants, showing that a point on the unit ball at distance at least ${\displaystyle 1/n}$ from the zero set of any degree-${\displaystyle n}$ polynomial always exists.^[13]