Discrete & Computational Geometry

, Volume 49, Issue 3, pp 478–484

# Combinatorial Generalizations of Jung’s Theorem

• Arseniy V. Akopyan
Article

## Abstract

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.

## Keywords

Jung’s theorem Helly’s theorem Covering by a ball

## 1 Introduction

The famous theorem of Jung states that any set with diameter $$1$$ in $$\mathbb R ^d$$ can be covered by the ball of radius $$R_d=\sqrt{\tfrac{d}{2(d+1)}}$$ (see [3]).

The proof of this theorem is based on Helly’s theorem:

### Theorem 1

(Helly’s theorem) Let $$\fancyscript{P}$$ be a family of convex compact sets in $$\mathbb R ^d$$ such that an intersection of any $$d+1$$ of them is not empty. Then the intersection of all of the sets from $$\fancyscript{P}$$ is not empty.

Helly’s theorem has many generalizations. Katchalski and Liu in 1979 [8] proved a “fractional” version of Helly’s theorem and Kalai in 1984 [7] gave the strongest version of it. In 1979, Lovász suggested a “colorful” version of Helly’s theorem. In this paper, we give analogous generalizations of Jung’s theorem.

## 2 The Fractinal Version of Jung’s Theorem

Let us recall the fractional version of Helly’s theorem.

### Theorem 2

(Katchalski and Liu [8]) For every $$d \ge 1$$ and every $$\alpha \in (0,1]$$ there exists a $$\beta = \beta (d,\alpha ) > 0$$ with the following property. Let $$\mathcal X _1,\mathcal X _2,\ldots ,\mathcal X _n$$ be convex sets in $$\mathbb R ^d$$ such that $$\bigcap \mathcal X _i \ne \emptyset$$ for at least $$\alpha C_n^{d+1}$$ index sets $$I \subseteq [n]$$ of size $$(d + 1)$$. Then there exists a point contained in at least $$\beta n$$ sets among $$\mathcal X _i$$.

The best possible value of $$\beta (d,\alpha )$$ is $$1 - (1 - \alpha )^{1/(d+1)}$$ [7] and, in particular, $$\beta \rightarrow 1$$ as $$\alpha \rightarrow 1$$.

Using this we can prove the fractional version of Jung’s theorem.

### Theorem 3

For every $$d \ge 1$$ and every $$\alpha \in (0,1]$$ there exists a $$\beta = \beta (d,\alpha ) > 0$$ with the following property. Let $$\mathcal V$$ be an $$n$$-point set in $$\mathbb R ^d$$ such that for at least $$\alpha C^2_n$$ of pairs $$\{x,y\}$$ ($$x, y \in \mathcal V$$) the distance between $$x$$ and $$y$$ is less than $$1$$. Then there exists a ball of radius $$R_d$$, which covers $$\beta n$$ points of $$\mathcal V$$. Moreover, $$\beta \rightarrow 1$$ as $$\alpha \rightarrow 1$$.

### Proof

Let us show that $$\beta >0$$ exists.

We construct a graph $$G$$ on points of $$V$$ as vertices. Two vertices of $$G$$ are connected if and only if the distance between the points is not greater than $$1$$.

Then the degree of some vertex $$v$$ of $$G$$ is not less than $$\alpha (n-1)$$. The ball with center at the point corresponding to $$v$$ and radius $$1$$ contains at least $$\alpha (n-1)+1$$ points of $$\mathcal V$$. Any ball of radius $$1$$ can be covered by $$c_d$$ balls of radius $$R_d$$, where $$c_d$$ is a constant depending only on the dimension $$d$$. Then one of these balls should cover more than $$\frac{\alpha (n-1)}{c_d}$$ points of the set $$\mathcal V$$. Therefore the statement of the theorem holds for $$\beta ={\alpha }/{2c_d}$$, because
\begin{aligned} \frac{\alpha }{2c_d}n<\frac{\alpha (n-1)}{c_d}. \end{aligned}
Note that this argument works for any radius $$R$$ not necessarily equal to $$R_d$$.
Let us show that $$\beta \rightarrow 1$$ as $$\alpha \rightarrow 1$$. Any pair of vertices belongs to $$C_{n-2}^{d-1}$$ subsets of $$d+1$$ vertices. Therefore, if there are at most $$(1-\alpha )C^2_n$$ empty edges in $$G$$, then the number of incomplete subgraphs on $$d+1$$ vertices is not greater than
\begin{aligned} (1-\alpha )C^2_nC_{n-2}^{d-1}=(1-\alpha )C_n^{d+1}C^2_{d+1}. \end{aligned}
We see that among the $$C_n^{d+1}$$ subgraphs of $$d+1$$ vertices at least $$\alpha ^{\prime }C_n^{d+1}$$ are full, where $$\alpha ^{\prime }=1-(1-\alpha )C^2_{d+1}$$.

Consider the family $$\fancyscript{B}$$ of $$n$$ balls of radius $$R_d$$ with centers at the points of $$\mathcal V$$. If the distances between the centers of some set of balls is no greater than $$1$$, then these balls have nonempty intersection (because Jung’s theorem implies that these centers can be covered by a ball of radius $$R_d$$). Therefore, among $$C^{d+1}_n$$ subsystems of $$d+1$$ balls from $$\fancyscript{B}$$, at least $$\alpha ^{\prime }C_n^{d+1}$$ have nonempty intersection. From fractional Helly’s theorem, it follows that there is a point which belongs to $$\beta (d,\alpha ^{\prime })n$$ balls. The ball with the center at this point and radius $$R_d$$ covers $$\beta n$$ points of $$\mathcal V$$. To conclude the proof, we note that $$\alpha ^{\prime }$$ tends to $$1$$, as $$\alpha \rightarrow 1$$. The result of Kalai now implies that $$\beta (d,\alpha ^{\prime })$$ also tends to $$1$$. $$\square$$

Note that using an approximation arguments it is possible to prove the same theorem for a measure.

## 3 Close Sets

We will use the following definition.

### Definition 1

We call two nonempty sets $$\mathcal V _1$$ and $$\mathcal V _2$$ close if for any points $$x\in \mathcal V _1$$ and $$y\in \mathcal V _2$$, the distance between $$x$$ and $$y$$ is not greater than $$1$$.

It is easy to see that if two close sets $$\mathcal V _1$$ and $$\mathcal V _2$$ are given, the diameter of each of them is not greater than $$2$$. Moreover, the following theorem holds.

### Theorem 4

The union of several pairwise close sets in  $$\mathbb R ^d$$  can be covered by a ball of radius $$1$$.

### Proof

Denote by $$\mathcal X$$ one of the sets and by $$\mathcal Y$$ union of other sets. Without loss of generality, we may assume that $$\mathcal X$$ and $$\mathcal Y$$ are convex closed sets, because the condition of the theorem also holds for $$\mathrm{cl}\,(\mathrm{conv}\mathcal X )$$ and $$\mathrm{cl}\,(\mathrm{conv}\mathcal Y )$$. If $$\mathcal X$$ and $$\mathcal Y$$ have nonempty intersection, then a unit ball with center at any point from the intersection covers $$\mathcal X$$ and $$\mathcal Y$$.

Suppose they do not intersect. Choose points $$x\in \mathcal X$$ and $$y \in \mathcal Y$$ so that the length of $$[x, y]$$ is minimal. Let $$m$$ be the midpoint of the segment $$[x,y]$$. We will show that the unit ball with center at $$m$$ covers $$\mathcal X$$ and $$\mathcal Y$$. It is known that the hyperplane perpendicular to $$[x,y]$$ and passing through $$m$$ separates $$\mathcal X$$ and $$\mathcal Y$$ (see [6]). Take any point $$z$$ from $$\mathcal X$$. Note that the angle $$zmy$$ is obtuse (Fig. 1). Therefore the segment $$[m, z]$$ is shorter than $$[y, z]$$, which is no longer than $$1$$. The same argument works for any point $$z$$ from $$\mathcal Y$$. $$\square$$

It is clear that the diameter of the covering ball in this theorem could not be decreased. The following two questions arise naturally.

Suppose that a family of pairwise close sets $$\mathcal V _1,\,\mathcal V _2$$, ..., $$\mathcal V _n$$ in $$\mathbb R ^d$$ is given.
• 1.  What is the minimal  $$R$$  so that at least one of the sets  $$\mathcal V _i$$  can be covered by a ball of radius $$R$$?

• 2.  What is the minimal $$D$$  so that at least one of the sets  $$\mathcal V _i$$ has diameter no greater than $$D$$?

For $$n=3$$ and $$d=2$$, the author learned the answer to the first question from Vladimir Dol’nikov. His arguments work well for all $$n>d$$. In Theorem 5, we find the exact value of $$R$$ for all pairs $$d$$ and $$n$$ in the first question. In Theorem 7, we show that the second question is equivalent to the well-known problem about spherical antipodal codes.

## 4 Colorful Jung’s Theorem

### Theorem 5

Let $$\mathcal V _1,\,\mathcal V _2$$, ..., $$\mathcal V _n$$ be pairwise close sets in $$\mathbb R ^d$$. Then one of the sets $$\mathcal V _i$$ can be covered by a ball of radius $$R$$.
\begin{aligned} R&= \tfrac{1}{\sqrt{2}}\quad {\textit{if}}\; n\le d;\\ R&= R_d=\sqrt{\tfrac{d}{2(d+1)}}\quad {\textit{if}} \;n > d. \end{aligned}

### Proof

First, let us show that $$R$$ from the statement of the Theorem is minimal. Suppose $$n \le d$$. Consider a crosspolytope with $$2n$$ vertices and with edge length $$1$$. Let the sets $$\mathcal V _i$$ be the pairs of opposite vertices of the crosspolytope. Then the distance between any two points from different sets is equal to $$1$$, and distance between two points from the same set is exactly $$\sqrt{2}$$. Therefore the radius of the minimal cover ball for each of the sets equals $$\tfrac{1}{\sqrt{2}}$$.

For $$n>d$$, let the sets $$\mathcal V _i$$ coincide with regular simplices with edge length $$1$$. In this case, the sets $$\mathcal V _i$$ are close to each other and the radius of the minimal cover ball for each set equals $$R_d$$.

Now let us show that one of the sets $$\mathcal V _i$$ can be covered by a ball of radius $$R$$. Without loss of generality, we may assume that the sets $$\mathcal V _i$$ are all closed, since the condition of the theorem holds for the closure of these sets.

Suppose $$n \le d$$. For any set $$\mathcal V _i$$, consider the minimal ball $$B(o_i, r_i)$$ covering this set. Let $$r_1$$ be the minimal radius among $$r_i$$. Note that there exists a point $$x$$ of the set $$\mathcal V _2$$ which does not belong to the interior of the ball $$B(o_1, r_1)$$. Indeed, if all the points of the set $$\mathcal V _2$$ belong to the interior of the ball $$B(o_1, r_1)$$, then $$\mathcal V _2$$ can be covered by a ball of radius less than $$r_1$$. This contradicts the assumption that $$r_1$$ is minimal among $$r_i$$. Let $$s$$ be a hyperplane passing through the point $$o_1$$ and perpendicular to the segment $$[o_1,x]$$ (Fig. 2). This hyperplane divides the sphere $$S(o_1, r_1)$$ onto two hemispheres. Denote by $$S^{\prime }(o_1, r_1)$$ the hemisphere, which lies in the halfspace generated by $$s$$ not containing $$x$$. We need the following lemma (see [3] statements 2.6 and 6.1 or [5] Lemma 2).

### Lemma 1

If a sphere $$S$$ is the boundary of the minimal ball which covers a closed set $$\mathcal V$$, then the center of $$S$$ belongs to $$\mathrm{conv}(\mathrm{cl}\,\mathcal V \cap S)$$.

Therefore the hemisphere $$S^{\prime }(o_1, r_1)$$ contains at least one point $$y$$ from the set $$\mathcal V _1$$. Note that distance from any point of the hemisphere $$S^{\prime }(o_1, r_1)$$ to $$x$$ is no less than $$\sqrt{2}r_1$$. Thus the distance between the points $$x$$ and $$y$$ is no less than $$\sqrt{2}r_1$$. Since $$\mathcal V _1$$ and $$\mathcal V _2$$ are close and the distance between $$x$$ and $$y$$ is no greater than $$1$$, we have $$r_1\le \frac{1}{\sqrt{2}}$$.

Consider the second case: $$n\ge d+1$$. This argument is due to V. L. Dol’nikov.

We will use the following theorem of Lovász [1].

### Theorem 6

(Colorful Helly’s theorem) Let $$\fancyscript{F}_1,\fancyscript{F}_2, \ldots ,\fancyscript{F}_{d+1}$$ be $$d+1$$ finite families of convex sets in $$\mathbb R ^d$$. If $$\cap _{i=1}^{d+1}\mathcal X _i \ne \emptyset$$ for all choices of $$\mathcal X _1 \in \fancyscript{F}_1,\,\mathcal X _2 \in \fancyscript{F}_2$$, ..., $$\mathcal X _{d+1} \in \fancyscript{F}_{d+1}$$, then for some $$i$$ the tersection of all sets from $$\fancyscript{F}_i$$ is not empty.

Let $$\fancyscript{B}_i$$ be a family of balls of radius $$R_d$$ with centers at the points of the set $$\mathcal V _i$$. Note that this set of families satisfies condition of colorful Helly’s theorem. Indeed, if we choose one ball from each of the sets $$\mathcal B _i\in \fancyscript{B}_i$$, then the distance between its centers will not be greater than $$1$$. From Jung’s theorem, it follows that they can be covered by a ball $$\mathcal B$$ of radius $$R_d$$. This means that the center of $$\mathcal B$$ is contained in all sets $$\mathcal B _i,i=1,2, \ldots ,n$$. Thus they have nonempty intersection.

Applying colorful Helly’s theorem to the families $$\fancyscript{B}_i$$, we obtain that for some $$i$$, the balls of the family $$\fancyscript{B}_i$$ have nonempty intersection. Therefore, all points of the set $$\mathcal V _i$$ can be covered by a ball of radius $$R_d$$ centered at any point of intersection of all balls from the family $$\fancyscript{B}_i$$. $$\square$$

### Remark 1

Note that, by the same arguments, it can be shown that for $$n \le d$$ all sets except one can be covered by a ball of radius $$1/\sqrt{2}$$. For the proof, one considers the minimal ball which covers all sets except one.

If $$n>d$$, then all but $$d$$ sets can be covered by a ball of radius $$R_d$$. Here one should use the modified version of colorful Helly’s theorem: Suppose a collection of families $$\{\fancyscript{F}_i\}$$ of convex sets is given and the intersection of any $$d+1$$ sets from different families is not empty. Then there is a point which pierces all but $$d$$ sets from the families $$\{\fancyscript{F}_i\}$$.

The proof is the same as the classical proof of colorful Helly’s theorem (see [9]): consider $$d$$ sets from different families whose intersection has the lowest higher point among all intersections of $$d$$ sets from different families. This point should be contained in all but the chosen $$d$$ sets of the families $$\{\fancyscript{F}_i\}$$.

## 5 The Bound on the Diameter of a Set

In this section, we study a bound on the diameter of $$\mathcal V _i$$.

### Definition 2

A spherical code is a finite set of points in $$\mathbb S ^d$$. A spherical code is called antipodal if it is symmetric with respect to the center of the sphere.

A spherical code $$\mathcal W$$ of $$n$$ points is called optimal if it has a maximal minimal diameter ($$\min |x-y|$$, where $$x,\,y \in \mathcal W$$ and $$x\ne y$$) among all spherical antipodal codes of the cardinality $$n$$.

By $$D_d(n)$$, we denote the minimal diameter of the optimal spherical antipodal code of cardinality $$2n$$ on the unit sphere $$\mathbb S ^{d-1}$$.

### Theorem 7

Let $$\mathcal V _1,\,\mathcal V _2$$, ..., $$\mathcal V _n$$ be pairwise close sets in $$\mathbb R ^d$$. Then one of the sets $$\mathcal V _i$$ has diameter no greater than
\begin{aligned} D=\frac{2}{\sqrt{4-D_d(n)^2}}. \end{aligned}

### Proof

Suppose otherwise. Then the diameter of each $$\mathcal V _i$$ is no less than $$D^{\prime }>D$$. Again, we may assume that all sets $$\mathcal V _i$$ are closed convex sets. Then in each set $$\mathcal V _i$$, it is possible to choose two points $$a_i$$ and $$b_i$$ with distance between them equal to $$D^{\prime }$$. So, we can assume that each set $$\mathcal V _i$$ is a two-point set: $$\mathcal V _i=\{a_i, b_i\}$$.

Let us show, without loss of generality, we may assume that the midpoints of the segments $$[a_i,b_i]$$ coincide.

Indeed, denote the origin by $$o$$.

Suppose
\begin{aligned} a_i^{\prime }=o+\frac{\overrightarrow{b_ia_i}}{2}, \quad b_i^{\prime }=o+\frac{\overrightarrow{a_ib_i}}{2}. \end{aligned}
Then
\begin{aligned}&|a_i^{\prime }-a_j^{\prime }|=|\overrightarrow{a_i^{\prime }a_j^{\prime }}|\\&\qquad \qquad \qquad =\frac{|\overrightarrow{a_ib_i}+\overrightarrow{b_ja_j}|}{2}=\frac{|\overrightarrow{a_ib_j}+\overrightarrow{a_jb_i}|}{2}\\&\qquad \qquad \qquad \le \tfrac{1}{2}(|a_i-b_j|+|a_j-b_i|) \le 1. \end{aligned}
The same inequality holds for all other pairs of points $$(a_i^{\prime }, b_j^{\prime })$$ and $$(b_i^{\prime }, b_j^{\prime })$$.

We may assume that the points of the sets $$\mathcal V _i$$ lie on the sphere of radius $$D^{\prime }/2$$ and form the antipodal code. Since the distance between the points, say, $$a_i$$ and $$a_j$$ is not greater than $$1$$, the distance between $$a_i$$ and $$b_j$$ should be greater than $$\sqrt{D^{\prime 2}-1}$$.

Therefore,
\begin{aligned} \begin{array}{l} \displaystyle \frac{2\sqrt{D^{\prime 2}-1}}{D^{\prime }}<D_d(n) \\ \quad \Leftrightarrow 4D^{\prime 2}-4<D_d(n)^2{D^{\prime 2}} \\ \quad \Leftrightarrow 4D^{\prime 2}-D_d(n)^2{D^{\prime 2}}<4 \\ \quad \Leftrightarrow D^{\prime 2}\le \displaystyle \frac{4}{4-D_d(n)^2}\\ \quad \Leftrightarrow D^{\prime }\le \displaystyle \frac{2}{\sqrt{4-D_d(n)^2}}=D. \end{array} \end{aligned}
This contradicts the assumption $$D^{\prime }>D$$ and concludes the proof. $$\square$$

It is clear that the corresponding optimal antipodal code yields equality for $$D$$.

Unfortunately, the precise value of $$D_d(n)$$ is known only for a few cases. In particular, the following numbers are known (see [2, 4]):
\begin{aligned} \begin{array}{l} \displaystyle D_2(n)=2\sin \displaystyle \frac{\pi }{2n};\\ D_d(n)=\sqrt{2} \text{ for} n\le d;\\ D_3(6)=\displaystyle \frac{2}{\sqrt{10 +2\sqrt{5}}};\\ D_4(12)=D_8(120)=D_{24}(98280)=1. \end{array} \end{aligned}

## Notes

### Acknowledgments

This research was partially supported by the Dynasty Foundation, the Presidents of Russian Federation Grant MD-352.2012.1, Russian Foundation for Basic Research Grants 11-01-00735 and 12-01-31281, the Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013, the Russian Government Project 11.G34.31.0053

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