Proof of L\'aszl\'o Fejes T\'oth's zone conjecture

TL;DR

This paper proves that the total width of any collection of zones covering the unit sphere must be at least π, resolving a long-standing question posed by Fejes Tóth in 1973.

Contribution

It provides a rigorous proof of Fejes Tóth's zone conjecture, establishing a lower bound on the total width of covering zones on the sphere.

Findings

Total width of covering zones on the sphere is at least π

Answer to Fejes Tóth's 1973 question

Completes the understanding of spherical zone coverings

Abstract

A zone of width $\omega$ on the unit sphere is the set of points within spherical distance $\omega/2$ of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.

Equations

$$\omega (F)\le \omega (C_1)+\omega (C_2)+\cdots +\omega (C_n).$$

$$\left|\mathbf{w}\right|\ge \sin \alpha \text{ and }\left|\mathbf{w}-{\mathbf{w}}_i\right|\le \sin \left(\alpha -{\alpha }_i\right)\text{ for all }i\in [n],$$

$$\left|\mathbf{w}\right|=\sin \alpha \text{ and }\left|\mathbf{w}-{\mathbf{w}}_i\right|=\sin \left(\alpha -{\alpha }_i\right)\text{ for some }i\in [n].$$

$$\begin{array}{r}{\left|\mathbf{w}-{\mathbf{w}}_i\right|}^2={\left|\mathbf{w}\right|}^2+{\left|{\mathbf{w}}_i\right|}^2-2\left|\mathbf{w}\right|\left|{\mathbf{w}}_i\right|\cos \angle (\mathbf{u},{\mathbf{u}}_i)>{\left|\mathbf{w}\right|}^2+{\sin }^2{\alpha }_i-2\left|\mathbf{w}\right|\sin {\alpha }_i\cos {\alpha }^{\prime }\\ ={\left(\left|\mathbf{w}\right|-\sin {\alpha }_i\cos {\alpha }^{\prime }\right)}^2+{\sin }^2{\alpha }_i{\sin }^2{\alpha }^{\prime }.\end{array}$$

$${\left(\sin \alpha -\sin {\alpha }_i\cos {\alpha }^{\prime }\right)}^2+{\sin }^2{\alpha }_i{\sin }^2{\alpha }^{\prime }={\cos }^2{\alpha }_i{\sin }^2{\alpha }^{\prime }+{\sin }^2{\alpha }_i{\sin }^2{\alpha }^{\prime }={\sin }^2{\alpha }^{\prime }.$$

$$\left|{\mathbf{w}}_1+{\mathbf{w}}_2\right|=\sin ({\alpha }_1+{\alpha }_2)\text{ and }\left|{\mathbf{w}}_1+{\mathbf{w}}_2+{\mathbf{w}}_3\right|=\sin \alpha .$$

$$\cos \angle ({\mathbf{w}}_{12},{\mathbf{w}}_1)=\frac{{\left|{\mathbf{w}}_{12}\right|}^2+{\left|{\mathbf{w}}_1\right|}^2-{\left|{\mathbf{w}}_2\right|}^2}{2\left|{\mathbf{w}}_{12}\right|\left|{\mathbf{w}}_1\right|}=\frac{{\sin }^2({\alpha }_1+{\alpha }_2)+{\sin }^2{\alpha }_1-{\sin }^2{\alpha }_2}{2\sin ({\alpha }_1+{\alpha }_2)\sin {\alpha }_1}=\cos {\alpha }_2.$$

$$\angle ({\mathbf{w}}_{12},{\mathbf{w}}_2)={\alpha }_1,\angle ({\mathbf{w}}_{123},{\mathbf{w}}_{12})={\alpha }_3,\angle ({\mathbf{w}}_{123},{\mathbf{w}}_3)={\alpha }_1+{\alpha }_2.$$

$${\delta }_1:=\angle ({\mathbf{w}}_{123},{\mathbf{w}}_{12})+\angle ({\mathbf{w}}_{12},{\mathbf{w}}_1)-\angle ({\mathbf{w}}_{123},{\mathbf{w}}_1)={\alpha }_2+{\alpha }_3-\angle ({\mathbf{w}}_{123},{\mathbf{w}}_1)>0.$$

$$\angle (\mathbf{w},{\mathbf{w}}_1)=\angle ({\mathbf{w}}_{123}+\epsilon {\mathbf{w}}_3,{\mathbf{w}}_1)=\angle ({\mathbf{w}}_{123},{\mathbf{w}}_1)+c_1\epsilon +O({\epsilon }^2)={\alpha }_2+{\alpha }_3-{\delta }_1+c_1\epsilon +O({\epsilon }^2).$$

$$\angle (\mathbf{w},{\mathbf{w}}_2)={\alpha }_1+{\alpha }_3-{\delta }_2+c_2\epsilon +O({\epsilon }^2).$$

$$\angle (\mathbf{w},{\mathbf{w}}_3)=\angle ({\mathbf{w}}_{123}+\epsilon {\mathbf{w}}_3,{\mathbf{w}}_3)=\angle ({\mathbf{w}}_{123},{\mathbf{w}}_3)-c_3\epsilon +O({\epsilon }^2)={\alpha }_1+{\alpha }_2-c_3\epsilon +O({\epsilon }^2),$$

$$L:=\left\{\sum _{i=1}^n{\epsilon }_i{\mathbf{w}}_i:({\epsilon }_1,\dots ,{\epsilon }_n)\in {\left\{\pm 1\right\}}^n\right\}.$$

$$\left|{\mathbf{w}}_1+\cdots +{\mathbf{w}}_i\right|=\sin ({\alpha }_1+\cdots +{\alpha }_i)\text{ for all }i\in [n].$$

Full text

1 Introduction

A plank (or slab, or strip) of width $w$ is a part of the $d$-dimensional Euclidean space $\mathbb{R}^{d}$ that lies between two parallel hyperplanes at distance $w$. Given a convex body $C$, its width is the smallest $w$ such that a plank of width $w$ covers $C$. In 1932, in the context of “degree of equivalence of polygons”, Tarski wrote in [Tar32]333See [MMS14, Chapter 7.3, 7.4] for English translations of [Moe32] and [Tar32]., and we quote from its English translation [MMS14, Chapter 7.4],

The width of the narrowest strip covering a plane figure $F$ we shall call the width of figure $F$, and we shall denote it by the symbol $\omega(F)$. …

A. If a figure $F$ contains in itself, as a part, a disk with diameter equal to the width of the figure (for example, if figure $F$ is a disk or a parallelogram) and if, moreover, we subdivide this figure into any $n$ parts $C_{1},C_{2},\ldots,C_{n}$, then

[TABLE]

Proof. For the case in which figure $F$ is a disk, the proof is an almost word-for-word repetition of the proof of lemma I in the cited article by Moese. For the general case, …

Here “the cited article by Moese” refers to [Moe32]11footnotemark: 1. For a long period of time, the authors of the current paper mistakenly overlooked the contribution of Moese.

The following conjecture, which is attributed to Tarski, seems to first appear in [Ban50].

Tarski’s plank problem. If a convex body of width $w$ is covered by a collection of planks in $\mathbb{R}^{d}$, then the total width of the planks is at least $w$.

It took almost twenty years before Bang proved Tarski’s conjecture in his memorable papers [Ban50, Ban51]. At the end of his paper, Bang asked whether his theorem could be strengthened by asking that the width of each plank should be measured relative to the width of the convex body being covered, in the direction normal to the plank. Although Bang’s conjecture still remains open, Ball [Bal91] established it for centrally symmetric convex bodies (see [Bal01] for the complex analogue). Ball also used Bang’s proof to improve the optimal density of lattice packings [Bal92].

Variants of Tarski’s plank problem continue to generate interest in the geometric and analytic aspects of coverings of a convex body (see [Bez13] for a recent survey). Most of these variants find its generalization in spherical geometry. Bezdek and Schneider [BS10] showed that the total inradius of the spherical convex domains444A convex spherical domain is a domain any two points of which can be joined by an arc of a great circle lying in the domain. The inradius of the domain $C$ is the spherical radius of the largest cap contained in $C$. covering the $n$-dimensional unit sphere is at least $\pi$ (see [AK12, Section 6] for a simplified proof by Akopyan and Karasev, and see [Kad05] for the Euclidean version by Kadets).

In this paper, we are concerned with a spherical analogue of a plank: a zone of width $\omega$ on the 2-dimensional unit sphere is defined as the set of points within spherical distance $\omega/2$ of a given great circle. In 1973, Fejes Tóth [Tót73] conjectured that if $n$ equal zones cover the sphere then their width is at least $\omega_{n}=\pi/n$. Rosta [Ros72] and Linhart [Lin74] proved the special case of 3 and 4 zones respectively. Lower bounds for $\omega_{n}$ were established by Fodor, Vígh and Zarnócz [FVZ16].

Fejes Tóth also formulated the generalized conjecture, which has been reiterated in [BMP05, Chapter 3.4, Conjecture 5] and the arXiv version of [AK12, Conjecture 8.3], for any set of zones (not necessarily of the same width) covering the unit sphere.

Fejes Tóth’s zone conjecture. The total width of any set of zones covering the sphere is at least $\pi$.

We completely resolve this conjecture and generalize it for the $d$-dimensional unit sphere $S^{d}$. Hereafter, all $d$-spheres are embedded in $\mathbb{R}^{d+1}$ and centered at the origin. For us, a great sphere is the intersection of a $d$-sphere and a hyperplane passing through the origin. This extends the notion of a great circle on a 2-dimensional sphere. Given a great sphere $C$, a zone $P$ of width $\omega$ on $S^{d}$ is defined as the set of points within spherical distance $\omega/2$ of $C$, and the central hyperplane of $P$ is the hyperplane through $C$.

Theorem 1**.**

The total width of any collection of zones $P_{1},\dots,P_{n}$ covering the unit $d$-sphere is at least $\pi$. For all $i\in[n]$, let $2\alpha_{i}$ be the width of $P_{i}$, and let $\ell_{i}$ be the line through the origin perpendicular to the central hyperplane of $P_{i}$. Equality holds if and only if after reordering the zones, $\ell_{1},\dots,\ell_{n}$ are coplanar lines in clockwise order such that the angle between $\ell_{i}$ and $\ell_{i+1}$ equals to $\alpha_{i}+\alpha_{i+1}$ for all $i\in[n]$ under the convention that $\ell_{n+1}=\ell_{1}$ and $\alpha_{n+1}=\alpha_{1}$.

Remark 1**.**

Note that this result does not follow from [BS10] because the zones are not convex in spherical geometry.

Our result is also connected to covering of a non-separable family of homothetic copies of a convex body. The first result in this direction was due to Goodman and Goodman [GG45]. They proved the following theorem conjectured by Erdős.

Theorem 2**.**

Let the circles with radii $r_{1},\dots,r_{n}$ lie in a plane. If no line divides the circles into two non-empty sets without intersecting any circle, then the given circles can be covered by a circle of radius $r=\sum_{i=1}^{n}{r_{i}}$.

Its potential spherical relative should concern caps: a cap of spherical radius $r$ on the unit $d$-sphere is defined as the set of points within spherical distance $r$ of a given point on the sphere. Recall that the projective duality in $\mathbb{R}^{d+1}$ interchanges lines through the origin with hyperplanes through the origin. By restricting to the $d$-sphere, we associate a great sphere to every point on the sphere. Thus a zone can be viewed as the dual of a cap. A result in reminiscence of Theorem 2 follows immediately from Theorem 1 and projective duality.

Corollary 3**.**

Let the caps with spherical radii $r_{1},\dots,r_{n}$ lie on the unit $d$-sphere. If every great sphere intersects at least one cap, then the total spherical radius $\sum_{i=1}^{n}r_{i}$ is at least $\pi/2$.

The proof of Theorem 1 will be given in the next section. In Section 3, we state a few more corollaries, one of which answers in the affirmative the following question of Fejes Tóth [Tót73] for the case of centrally symmetric domains:

Fejes Tóth’s zone problem. Is it true that if a convex spherical domain is covered with a set of zones of total width $w$, then it can be covered with one zone of width $w$?

We conclude, in Section 4, with some brief remarks.

2 Proof of Theorem 1

Our proof is inspired by Bang [Ban51], Goodman and Goodman [GG45]. Roughly speaking, the idea of the proof is to show that

Either a particular discrete subset not fully covered by the convex hulls (taken in $\mathbb{R}^{d+1}$) of the zones is contained in the unit ball; 2. 2.

Or several zones can be covered and replaced by one zone without increasing the total width.

In this section, we denote points by vectors leading to them from the origin. We need the following technical lemmas, the first of which is in the spirit of Theorem 2.

Lemma 4**.**

Suppose the total spherical radius of a collection of caps $D_{1},\dots,D_{n}$ on $S^{d}$ is at most $\pi/2$. For all $i\in[n]$, let $\alpha_{i}$ be the spherical radius of $D_{i}$, let $\mathbf{u}_{i}$ be the center of $D_{i}$, and set $\mathbf{w}_{i}:=\sin\alpha_{i}\mathbf{u}_{i}$. Denote $\mathbf{w}:=\sum_{i=1}^{n}\mathbf{w}_{i}$ and $\alpha:=\sum_{i=1}^{n}\alpha_{i}$. If

[TABLE]

then a cap of spherical radius $\leq\alpha$ can cover $\cup_{i=1}^{n}D_{i}$, and moreover, a cap of spherical radius $<\alpha$ can cover $\cup_{i=1}^{n}D_{i}$ except when

[TABLE]

Remark 2**.**

In case that (2) holds, by the law of sines, $\angle(\mathbf{w},\mathbf{w}_{i})=\alpha-\alpha_{i}$ (see Figure 1).

Proof.

We claim that the cap $D$ centered at $\mathbf{u}:=\mathbf{w}/\left\lvert{\mathbf{w}}\right\rvert$ of spherical radius $\alpha$ covers $\cup_{i=1}^{n}D_{i}$. Suppose on the contrary that say $D_{i}$ is not covered by $D$. Clearly, the angle between $\mathbf{u}$ and $\mathbf{u}_{i}$ is strictly greater than $\alpha^{\prime}:=\alpha-\alpha_{i}$. By the law of cosines and the monotonicity of cosine on $(0,\pi)$, we get

[TABLE]

Notice that $\left\lvert{\mathbf{w}}\right\rvert\geq\sin\alpha=\sin\alpha_{i}\cos\alpha^{\prime}+\cos\alpha_{i}\sin\alpha^{\prime}>\sin\alpha_{i}\cos\alpha^{\prime}$. The right hand side of (3) is at least

[TABLE]

Therefore $\left\lvert{\mathbf{w}-\mathbf{w}_{i}}\right\rvert>\sin\alpha^{\prime}=\sin(\alpha-\alpha_{i})$ contradicts (1).

Clearly if $\mathbf{w}>\sin\alpha$ or $\left\lvert{\mathbf{w}-\mathbf{w}_{i}}\right\rvert<\sin(\alpha-\alpha_{i})$, then the cap centered at $\mathbf{u}$ of spherical radius $\alpha-\varepsilon_{i}$ covers $D_{i}$ for some $\varepsilon_{i}>0$. Hence if $\left\lvert{\mathbf{w}}\right\rvert>\sin\alpha$ or $\left\lvert{\mathbf{w}-\mathbf{w}_{i}}\right\rvert<\sin(\alpha-\alpha_{i})$ for all $i\in[n]$, then the cap centered at $\mathbf{u}$ of spherical radius $\alpha-\min\left\{\varepsilon_{i}:i\in[n]\right\}$ covers $\cup_{i=1}^{n}D_{i}$. ∎

Lemma 5**.**

Suppose the total spherical radius of 3 caps $D_{1},D_{2},D_{3}$ is $\alpha\leq\pi/2$. For all $i\in[3]$, let $\alpha_{i}$ be the spherical radius of $D_{i}$, let $\mathbf{u}_{i}$ be the center of $D_{i}$, and set $\mathbf{w}_{i}:=\sin\alpha_{i}\mathbf{u}_{i}$. Suppose that

[TABLE]

If $\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}$ are not coplanar vectors, then a cap of spherical radius $<\alpha$ can cover $D_{1}\cup D_{2}\cup D_{3}$.

Proof.

Denote $\mathbf{w}_{12}:=\mathbf{w}_{1}+\mathbf{w}_{2}$ and $\mathbf{w}_{123}:=\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}$. By the law of cosines and trigonometric identities, we obtain that

[TABLE]

Therefore $\angle(\mathbf{w}_{12},\mathbf{w}_{1})=\alpha_{2}$. Similarly, one can compute

[TABLE]

Suppose that $\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}$ are not coplanar. Set $\mathbf{w}=\mathbf{w}(\varepsilon):=\mathbf{w}_{123}+\varepsilon\mathbf{w}_{3}$. We claim that the cap $D=D(\varepsilon)$ centered at $\mathbf{w}/\left\lvert{\mathbf{w}}\right\rvert$ of spherical radius $<\alpha$ can cover $D_{1}\cup D_{2}\cup D_{3}$ for some $\varepsilon>0$. Because $\mathbf{w}_{1},\mathbf{w}_{12},\mathbf{w}_{123}$ are not coplanar, by the spherical triangle inequality, we obtain

[TABLE]

Therefore there exists $c_{1}\in\mathbb{R}$ such that

[TABLE]

Similarly, $\delta_{2}:=\alpha_{1}+\alpha_{3}-\angle(\mathbf{w}_{123},\mathbf{w}_{2})>0$ and there exists $c_{2}\in\mathbb{R}$ such that

[TABLE]

Since $\mathbf{w}_{123},\mathbf{w}_{12},\mathbf{w}_{3}$ are coplanar, but not collinear, there exists $c_{3}>0$ such that

[TABLE]

In (4, 5, 6), one can take sufficiently small $\varepsilon>0$ so that $\angle(\mathbf{w},\mathbf{w}_{i})<\alpha-\alpha_{i}$ for all $i\in[3]$. This proves the claim. ∎

Remark 3**.**

Because $\angle(\mathbf{w}_{123},\mathbf{w}_{1})<\alpha_{2}+\alpha_{3},\angle(\mathbf{w}_{123},\mathbf{w}_{2})<\alpha_{1}+\alpha_{3}$ and $\angle(\mathbf{w}_{123},\mathbf{w}_{3})=\alpha_{1}+\alpha_{2}$, the cap, say $D_{123}$, centered at $\mathbf{w}_{123}/\left\lvert{\mathbf{w}_{123}}\right\rvert$ of spherical radius $\alpha=\alpha_{1}+\alpha_{2}+\alpha_{3}$ covers $\cup_{i=1}^{3}D_{i}$. The idea of the proof of Lemma 5 is to move the center of $D_{123}$ towards $\mathbf{w}_{3}/\left\lvert{\mathbf{w}_{3}}\right\rvert$ slightly and shrink its spherical radius properly so that the cap would still cover $\cup_{i=1}^{3}D_{i}$ (see Figure 2).

Proof of Theorem 1.

Suppose that the total width of a collection of zones $P_{1},\dots,P_{n}$ on $S^{d}$ is less than $\pi$. Let $2\alpha_{i}$ be the width of $P_{i}$, let $H_{i}$ be the central hyperplane of $P_{i}$, and let $\mathbf{u}_{i}$ be a unit normal vector of $H_{i}$. Set $\mathbf{w}_{i}:=\sin\alpha_{i}\mathbf{u}_{i}$ for all $i\in[n]$. Consider the set

[TABLE]

Let $\mathbf{w}:=\sum_{i=1}^{n}\varepsilon_{i}\mathbf{w}_{i}$ achieve the maximal norm among points of $L$. For every $i\in[n]$, using the fact that $\left\lvert{\mathbf{w}}\right\rvert\geq\left\lvert{\mathbf{w}-2\varepsilon_{i}\mathbf{w}_{i}}\right\rvert$, one can easily see that the distance from $\mathbf{w}$ to $H_{i}$ is at least $\left\lvert{\mathbf{w}_{i}}\right\rvert=\sin\alpha_{i}$ (see Figure 3).

If $\left\lvert{\mathbf{w}}\right\rvert<1$, then $\mathbf{w}/\left\lvert{\mathbf{w}}\right\rvert\notin\cup_{i=1}^{n}P_{i}$. Otherwise, $\left\lvert{\mathbf{w}}\right\rvert\geq 1$. Without loss of generality, we may assume that $\varepsilon_{1}=\dots=\varepsilon_{n}=1$. In particular, $\left\lvert{\sum_{i=1}^{n}\mathbf{w}_{i}}\right\rvert>\sin\left(\sum_{i=1}^{n}\alpha_{i}\right)$ because the total width $2\alpha_{1}+\dots+2\alpha_{n}<\pi$. Find the minimal set $I\subset[n]$ such that $\left\lvert{\sum_{i\in I}\mathbf{w}_{i}}\right\rvert>\sin\left(\sum_{i\in I}\alpha_{i}\right)$. For each $i\in I$, denote by $D_{i}$ the cap centered at $\mathbf{u}_{i}$ of spherical radius $\alpha_{i}$. By the minimality of $I$, we can apply the first part of Lemma 4 to $\left\{D_{i}:i\in I\right\}$ to find a cap, say $D$, covering $\cup_{i\in I}D_{i}$ of spherical radius $\leq\sum_{i\in I}\alpha_{i}$. Using the projective duality, one can check that the dual of $D_{i}$ is $P_{i}$ and the dual of $D$ is a zone, say $P$, of width $\leq\sum_{i\in I}\alpha_{i}$. Moreover, because $\cup_{i\in I}D_{i}$ is covered by $D$ and the projective duality preserves containment, $\cup_{i\in I}P_{i}$ is covered by $P$. Clearly, $\left\lvert{I}\right\rvert>1$. Now we can reduce the number of zones by replacing $\left\{P_{i}:i\in I\right\}$ by $P$ without increasing the total width, and we repeat the above argument for zones $\left\{P\right\}\cup\left\{P_{i}:i\in[n]\setminus I\right\}$.

Finally, we investigate the case when equality holds. Suppose that zones $P_{1},\dots,P_{n}$ of total width $2\alpha=\pi$ cover $S^{d}$. For all $i\in[n]$, let $D_{i}$ be the dual cap555Strictly speaking, the dual of a zone $P$ is a pair of equal caps. However, if a normal vector $\mathbf{u}$ of the central hyperplane $H$ of $P$ is chosen, then the dual cap of $P$ is defined as one of these caps in the half-space determined by $H$ containing $\mathbf{u}$. of $P_{i}$. From the previous argument, we must have $\left\lvert{\mathbf{w}}\right\rvert=1=\sin\alpha$. Moreover, for any $I\subset[n]$, caps $\left\{D_{i}:i\in I\right\}$ cannot be covered by a cap of spherical radius $<\sum_{i\in I}\alpha_{i}$ since otherwise we can cover the sphere with zones of total width $<\pi$.

The second part of Lemma 4 shows that $\left\lvert{\mathbf{w}-\mathbf{w}_{i}}\right\rvert=\sin(\alpha-\alpha_{i})$ for some $i\in[n]$. Without loss of generality, assume that $\left\lvert{\mathbf{w}-\mathbf{w}_{n}}\right\rvert=\sin(\alpha-\alpha_{n})$. Again, apply the second part of Lemma 4 to $D_{1},\dots,D_{n-1}$, it must be the case that $\left\lvert{\mathbf{w}-\mathbf{w}_{n}-\mathbf{w}_{i}}\right\rvert=\sin(\alpha-\alpha_{n}-\alpha_{i})$ for some $i\in[n-1]$. After repeating this argument for $(n-1)$ times, we can assume, without loss of generality, that

[TABLE]

Since $\left\lvert{\mathbf{w}_{1}+\mathbf{w}_{2}}\right\rvert=\sin(\alpha_{1}+\alpha_{2})$, $\left\lvert{\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}}\right\rvert=\sin(\alpha_{1}+\alpha_{2}+\alpha_{3})$ and $D_{1},D_{2},D_{3}$ cannot be covered by a cap of spherical radius $<\alpha_{1}+\alpha_{2}+\alpha_{3}$, Lemma 5 implies that $\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}$ are coplanar. Now let $\mathbf{w}_{2}^{\prime}:=\mathbf{w}_{1}+\mathbf{w}_{2}$ and $D_{2}^{\prime}$ be the cap centered at $\mathbf{w}_{2}^{\prime}/\left\lvert{\mathbf{w}_{2}^{\prime}}\right\rvert$ of spherical radius $\alpha_{2}^{\prime}:=\alpha_{1}+\alpha_{2}$. One can check that $D_{2}^{\prime}$ cover $D_{1},D_{2}$, and so $D_{2}^{\prime},D_{3},D_{4}$ cannot be covered by a cap with radius $<\alpha_{2}^{\prime}+\alpha_{3}+\alpha_{4}$. Since, in addition, $\left\lvert{\mathbf{w}_{2}^{\prime}+\mathbf{w}_{3}}\right\rvert=\sin(\alpha_{2}^{\prime}+\alpha_{3})$ and $\left\lvert{\mathbf{w}_{2}^{\prime}+\mathbf{w}_{3}+\mathbf{w}_{4}}\right\rvert=\sin(\alpha_{2}^{\prime}+\alpha_{3}+\alpha_{4})$, Lemma 5 implies that $\mathbf{w}_{2}^{\prime},\mathbf{w}_{3},\mathbf{w}_{4}$ are coplanar, hence $\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3},\mathbf{w}_{4}$ are coplanar. Repeating this argument, we obtain that $\mathbf{w}_{1},\dots,\mathbf{w}_{n}$ are coplanar. This reduces the problem for $S^{d}$ to $S^{1}$, and it is easy to check the equality condition for $S^{1}$. ∎

Remark 4**.**

Alternatively, to find a point of $L$ that is not covered with the convex hulls of the zones, one could use Bang’s approach [Ban51] by optimizing a cleverly chosen quadratic function on $\left\{\pm 1\right\}^{n}$. Our proof makes use of the neat trick, due to Bognár [Bog61], by picking the vector achieving the maximal norm among points of $L$.

3 Other corollaries

The following two corollaries follow easily from Theorem 1. We call a pair of equal caps an antipodal cap if they are antipodal to each other.

Corollary 6**.**

Given a family of antipodal caps of radii $r_{1},\dots,r_{n}$ on the unit $d$-sphere, it is always possible to find a point common to them if the total spherical radius $\sum_{i=1}^{n}r_{i}$ is at least $(n-1)\frac{\pi}{2}$.

Proof.

Apply Theorem 1 to the complement of the antipodal caps. ∎

Corollary 7**.**

Given a family of great spheres $C_{1},\dots,C_{n}$ on the unit $d$-sphere, it is always possible to find an open cap of spherical radius $\pi/2n$ not intersecting any $C_{i}$.

Proof.

Apply Theorem 1 to the zones of width $\pi/n$ with central great spheres $C_{1},\dots,C_{n}$. ∎

A spherical domain $W$ is centrally symmetric with respect to a point $c$ if for every $a\in W$ there is $b\in W$ such that $c$ is the middle point of the shortest arc connecting $a$ and $b$. The following corollary partially confirms Fejes Tóth’s zone problem.

Corollary 8**.**

If a centrally symmetric convex spherical domain is covered with a set of zones of total width $w$, then it can be covered with one zone of width $w$. In particular, the total width of zones covering a cap of spherical radius $r$ is at least $2r$.

Proof.

Suppose a convex spherical domain $W$ is centrally symmetric with respect to $c$, and is covered with a set of zones $P_{1},\dots,P_{n}$ of total width $w$. Suppose $D$ is the largest cap contained in $W$ and centered at $c$. Let $r$ be the spherical radius of $D$. Let $D^{\prime}$ be the equal cap that is antipodal to $D$. The complement of $D\cup D^{\prime}$ is a zone of width $\pi-2r$, which together with $P_{1},\dots,P_{n}$ covers the sphere. Theorem 1 says that $\pi-2r+w\geq\pi$ or equivalently $2r\leq w$. Let $D$ touch the boundary of $W$ at $a,b$ which are centrally symmetric with respect to $c$. One can check that the zone of width $2r$ with central great circle passing through $c$ and perpendicular to arc $ab$ covers $W$. ∎

4 Concluding remarks

Besides Fejes Tóth’s zone problem at the end of Section 1, let us mention four more conjectures related to the theme of the article. We believe the following strengthening of Theorem 1 holds.

Conjecture 1**.**

A spherical segment is the solid defined by cutting the unit ball with a pair of parallel planes. Its width is the length of the shortest arc on the sphere whose endpoints touch both parallel planes. If the unit ball is covered with a collection of spherical segments, then the total width of the spherical segments is at least $\pi$.

We believe that an even stronger version holds.

Conjecture 2**.**

If the unit sphere is covered with a collection of spherical segments, then the total width of the spherical segments is at least $\pi$.

In connection with the spherical relative of Theorem 2, we suggest the following conjecture.

Conjecture 3**.**

Let the caps with spherical radii $r_{1},\dots,r_{n}$ lie on a hemisphere of the unit $d$-sphere. If no great sphere divides the caps into two non-empty sets without intersecting any cap, then the given caps can be covered by a cap of spherical radius $r=\sum_{i=1}^{n}r_{i}$.

Remark 5**.**

The converse of Corollary 3 is true up to rearrangement of the given caps in the following sense. If the total spherical radius of the caps is at least $\pi/2$, one can align their centers along a great sphere so that every great sphere intersects at least one cap. Now fix such a family of caps of total spherical radius $\pi/2$. Also choose an arbitrary $\pi/3$-covering $X$ for the $d$-sphere, that is, a finite subset of the $d$-sphere such that every point on the $d$-sphere is within spherical distance $\pi/3$ of some point in $X$. For each point in $X$, we add to the family a tiny cap centered at that point. Thus we obtain a non-separable family of caps of total spherical radius slightly larger than $\pi/2$. However, just to cover $X$, we need a cap of spherical radius $\geq 2\pi/3$. This construction shows the assumption that the given caps lie on a hemisphere is indispensable to Conjecture 3.

Informally, the proof of Theorem 1 takes advantage Bang’s ingenious approach [Ban51] — we know a bit more about the zones in case Bang’s argument does not apply. Moreover, Theorem 1 can be seen as a strengthening of Tarski’s plank theorem [Tar32]. The following conjecture of Bezdek [Bez03] shares these features.

Conjecture 4** (Bezdek).**

Take a unit disk and turn it into an annulus by cutting a sufficiently small concentric circular hole. If the annulus is covered with a collection of planks, then the total width of the planks is at least $1$.

We mention a conjecture attributed to Bezdek and Schneider [AK12, Problem 6.2]. It was solved for $r\geq\pi/2$ [BS10, Theorem 6.1], and it is closely related to Corollary 8.

Conjecture 5** (Bezdek–Schneider).**

If a cap of spherical radius $r$ is covered with a collection of convex spherical domains, then the total inradius of the domains is at least $r$.

Another interesting direction of generalization is known as the fractional plank problem (see [AHKM02] and [AKP14, Section 7] for results and conjectures). Finally, we refer the readers to [BMP05, Chapter 3.4] for a more comprehensive list of open problems related to Tarski’s plank problem.

Acknowledgements

We would like to thank Arseniy Akopyan and Roman Karasev for inspirational discussions on Fejes Tóth’s zone conjecture, and for bringing the historical account of Tarski’s plank problem to our attention. We are grateful to Ron Aharoni and Benjamin Matschke for discussions on related topics. We also benefited from the constructive comments of the anonymous referee.