### Spherical Tarski's Plank Problem

Arizona State University
July 21, 2023
Joint work with Саша Полянский (Alexandr Polyanskii)

### 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 $C$.

Alfred Tarski in 1932 proved for disks

Thøger Bang in 1950 proved for convex bodies

### Zone Zone of width $\omega$ is part of unit 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.

Appeared in Research Problems in Discrete Geometry

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 $100w \ge 2.032$;  2017 J.–Polyanskii: any set of zones, any dimension, charaterize the equality cases.

### Plan of talk

Proofs of Tarski's plank problem

Proof of Fejes Tóth's zone conjecture

### Tarski's proof

Total width of planks covering unit disk is $\ge 2$. Planks cover disk$\implies$ Arches cover hemisphere$\implies \sum \pi w_i \ge 2\pi$

Tarski's proof applies to  $2 \times$ inradius of $C$ = width of $C$,

but not to ### Bang's proof

$\vec{w}_i :=$ "direction" of plank $i$

$L := \{\pm \vec{w}_1 \pm \dots \pm \vec{w}_n\}$

Idea 1: $L$ cannot be covered by the planks.

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

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

In general, plank $i$: $|\vec{w} \cdot \vec{w}_i$$+ b_i$$| \le |\vec{w}_i|^2$

$\sum \epsilon_i\epsilon_j (\vec{w}_i\cdot\vec{w}_j)$ $+\sum b_i\epsilon_i$

### Plan of Talk

Proofs of Tarski's plank problem

Proof of Fejes Tóth's zone conjecture

### Fejes Tóth's zone conjecture $\vec{w}_i :=$ direction of zone $i$.

Bang says: 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 ...

Maybe 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 $|\vec{w}_1 + \vec{w}_2| \ge \sin(\alpha_1 + \alpha_2)$.

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

### Punchline

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!

### Goodman–Goodman Theorem

Let disks with radii $r_1, \dots, r_n$ lie in plane.
If no line "seperates" disks,
then they can be coverered by disk of radius $\sum_i r_i$.

Idea: Let $\vec{x}_i$ be center of disk $i$. Consider the disk centered at $\sum_i r_i\vec{x}_i / \sum_i r_i$ with radius $\sum_i r_i$.

Inspiration: Use zone with direction $\vec{w}$ to replace.

### 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 ...$\ge 2r$

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

### Bang's plank conjecture

Measure relative to $C$ in direction normal to plank. Conjecture: total relative width of planks covering $C$ is at least $1$.

Keith Ball in 1991 proved for symmetric bodies.

### Bezdek's annulus problem Conjecture: total width of planks covering punctured disk is at least diameter. Arizona State University
zilinj@mit.edu