Plank

Plank

A plank (or slab, strip) of width $w$ is part of $\mathbb{R}^d$ that lies between two parallel hyperplanes at distance $w$.

Tarski's plank problem

The width of $C$ is the smallest width of plank covers $C$.

If convex body $C$ is covered by planks, then ...

total width of planks is at least width of $C$.

Proved by Thøger Bang in 1950.

Zone

A zone of width $\omega$ is part of sphere that lies within spherical distance $\omega/2$ of a given great circle.

Fejes Tóth's zone conjecture

The total width of zones covering sphere is at least ... $\pi$.

Research Problems: Exploring a Planet.
American Mathematical Monthly, 1973.

1972 Rosta: 3 zones of equal width;

1974 Linhart: 4 zones of equal width;

2016 Fodor, Vígh and Zarnócz: if $100$ zones of equal width $w$ cover sphere, then $w \ge 0.02032$;

2017 J.–Polyanskii: any set of zones, any dimension, charaterize the equality cases.

Bang's proof

$\vec{w}_i :=$ vector representing plank $i$.

$L := \{\pm \vec{w}_1 \pm \dots \pm \vec{w}_n\}$ = projection of $\{\pm 1\}^n$.

Idea 2: $L$ can be embeded in $C$ of large width.

Bognár's simplification

Special case: all planks are centered at O.

Claim: $\vec{w}\in L$ achieving max norm is not covered.

$|\vec{w}| \ge |\vec{w} \pm 2\vec{w}_i| \implies w$ not covered by plank $i$.

General case

$\vec{w} = \pm \vec{w}_1 \pm \dots \pm \vec{w}_n \in L$ achieves max norm iff

$|\epsilon_1 \vec{w}_1 + \dots + \epsilon_n \vec{w}_n|^2$ is maximized on $\{\pm 1\}^n$.

Fejes Tóth's zone conjecture

Suppose $\vec{w}_i$ is vector representing zone $i$.

Bang: some $\vec{w} = $$\epsilon_1$$\vec{w}_1 + \dots +$$\epsilon_n$$\vec{w}_n$ is not covered.

To Bang or to ...?

If $|\vec{w}| \le 1$, then $\hat{w}$ is not covered.

Otherwise, $\vec{w} = \vec{w}_1 + \dots + \vec{w}_n$ is of big maginitude ...

we can merge some zones!

When to merge?

$\angle(\vec{w}_1, \vec{w}_2) \le \alpha_1 + \alpha_2$

Trignometry

$\angle(\vec{w}_1, \vec{w}_2) \le \alpha_1 + \alpha_2$

$\cos \angle(\vec{w}_1, \vec{w}_2) \ge \cos(\alpha_1 + \alpha_2)$

$|\vec{w}_1 + \vec{w}_2|^2 = |\vec{w}_1|^2 + 2|\vec{w}_1||\vec{w}_2|\cos\angle(\vec{w}_1, \vec{w}_2) + |\vec{w}_2|^2$

$\ge \sin^2\alpha_1 + 2\sin\alpha_1\sin\alpha_2\cos(\alpha_1 + \alpha_2) + \sin^2\alpha_2$

$= \dots = \sin(\alpha_1 + \alpha_2)^2$.

Can merge 2 zones when $|\vec{w}_1 + \vec{w}_2| \ge \sin(\alpha_1 + \alpha_2)$.

Punchline

In general, can merge some zones when $|\vec{w}_1 + \dots + \vec{w}_n| \ge \sin(\alpha_1 + \dots + \alpha_n)$.

Assume half of total width $\alpha_1 + \dots + \alpha_n < \pi / 2$.

If $|\vec{w}| \leq 1 $, $\hat{w}$ is not covered.

Otherwise $|\vec{w}| > 1 > \sin(\alpha_1 + \dots + \alpha_n)$, merge!

Application

Projective duality

Great circle $\leftrightarrow$ Antipodal points

Zone $\leftrightarrow$ Antipodal caps

If every great circle intersects (antipodal) caps...

then total radius of caps is at least $\pi/2$.

Covering a cap

The total width of zones covering cap of radius $r$...

is at least $2r$.

Conjecture (Fejes Tóth): ... covering spherical convex domain $D$ is at least width of $D$.

Open problem

Bang's plank conjecture

Measure relative to $C$ in direction normal to plank.

Conjecture: total relative width of planks covering $C$ is at least $1$.

Bezdek's annulus problem

Conjecture: total width of planks covering punctured disk is at least diameter.