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# Covering functionals of cones and double cones

*Journal of Inequalities and Applications*
**volume 2018**, Article number: 186 (2018)

## Abstract

The least positive number *γ* such that a convex body *K* can be covered by *m* translates of *γK* is called the covering functional of *K* (with respect to *m*), and it is denoted by \(\Gamma_{m}(K)\). Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s quantitative program for attacking Hadwiger’s covering conjecture. Estimations of covering functionals of cones and double cones, which are best possible for certain pairs of *m* and *K*, are presented.

## 1 Introduction

A compact convex set \(K\subseteq\mathbb{R}^{n}\) having interior points is called a *convex body*. The *interior* and *boundary* of *K* are denoted by int*K* and bd*K*, respectively. We write \(\mathcal{K}^{n}\) for the set of convex bodies in \(\mathbb{R}^{n}\). Concerning the least number \(c(K)\) of translates of int*K* needed to cover a convex body *K*, there is a long-standing conjecture:

### Conjecture 1

(Hadwiger’s covering conjecture)

*For each*
\(K\in\mathcal{K}^{n}\), \(c(K)\)
*is bounded from above by*
\(2^{n}\), *and this upper bound is attained only by parallelotopes*.

We refer to [1–4], and [5] for more information and references about this conjecture. Note that, for each \(K\in\mathcal{K}^{n}\), \(c(K)\) equals the least number of smaller homothetic copies of *K* needed to cover *K* (see, e.g., [1, p. 262, Theorem 34.3]). Therefore, \(c(K)\leq m\) for some \(m\in \mathbb{Z}^{+}\) if and only if \(\Gamma_{m}(K)<1\), where \(\Gamma_{m}(K)\) is defined by

and called *the covering functional of*
*K*
*with respect to*
*m*. A closely related concept is studied in functional analysis. Given a bounded subset *M* of a normed space *E* with unit ball *B*, the *m-th entropy number*
\(\varepsilon_{m}(M)\) of *M* is defined by (cf. [6, p. 6–7])

When *K* is a translate of the unit ball *B* of a finite-dimensional normed space *E*, we have \(\Gamma_{m}(K)=\varepsilon_{m}(K)\).

Clearly, for each \(m\in\mathbb{Z}^{+}\), \(\Gamma_{m}(\cdot)\) is affinely invariant. More precisely,

where \(\mathcal{A}^{n}\) is the set of non-degenerate affine transformations from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\). Thus we identify convex bodies that are affinely equivalent, and when writing \(\mathcal{K}^{n}\) we are actually referring to the quotient space of \(\mathcal{K}^{n}\) with respect to affine equivalence.

For each pair \(K_{1}\), \(K_{2}\) of convex bodies in \(\mathcal{K}^{n}\), the *Banach–Mazur distance*
\(d_{\mathrm{BM}}(K_{1},K_{2})\) (also called the *Asplund metric*, cf. [7]) between them is defined by

Then \((\mathcal{K}^{n},d_{\mathrm{BM}})\) is a compact metric space (cf. [8] and [7]). Zong (cf. [9]) proved that \(\Gamma _{m}(\cdot)\) is uniformly continuous on \(\mathcal{K}^{n}\). Bezdek and Khan improved this result by showing that \(\Gamma_{m}(\cdot)\) is Lipschitz continuous on \(\mathcal{K}^{n}\) with \((n^{2}-1)/(2\ln n)\) as a Lipschitz constant (cf. [10]). These results show that each \(K\in\mathcal{K}^{n}\) can be covered by at most \(2^{n}\) smaller homothetic copies of *K* if and only if

Due to these facts, estimating covering functionals of convex bodies is an important part of Zong’s quantitative program for attacking Hadwiger’s covering conjecture (cf. [9] for more details).

Our starting point is Theorem 1 in [9]: \(\Gamma_{8}(C)\leq \frac{2}{3}\) when *C* is a three-dimensional convex cone (the convex hull of the union of a planar convex body *K* and a singleton not contained in the plane containing *K*). By applying new ideas, we show that this estimation can be improved when \(\Gamma_{7}(K)<\frac{1}{2}\), and that this estimation can be extended to higher dimensional situations. Moreover, we also obtain an estimation of \(\Gamma_{m}(C)\) when *C* is a double cone.

## 2 Covering functionals of convex cones

We start with an elementary lemma.

### Lemma 1

*Let*
\(K\subset\mathbb{R}^{n}\)
*be a convex set*, \(x\in\mathbb{R}^{n}\), *and*
\(\lambda,\gamma\in [{0},{1} ]\). *Then*

### Proof

Let *z* be an arbitrary point in \((x+\gamma K)\cap K\). Then \(z-x\in \gamma K\) and \(z\in K\). Therefore

It follows that \(z\in\lambda x+(\lambda\gamma+1-\lambda)K\). □

For each \(\bar{x}\in\mathbb{R}^{n}\), we put \(x=(\bar{x},0)\in \mathbb{R}^{n+1}\). Each point \(x\in\mathbb{R}^{n+1}\) can be written in the form \((\bar{x},\alpha )\), where \(\bar{x}\in\mathbb{R}^{n}\) and \(\alpha\in\mathbb{R}\). Let *K* be a convex body in \(\mathcal{K}^{n}\) containing the *origin*
*o* in its interior, and *p* be a point in \(\mathbb{R}^{n+1}\setminus\mathbb{R}^{n}\times\{0\}\). Put \(C=\operatorname{conv}((K\times\{0\})\cup\{p\})\), i.e., *C* is the *cone* having *p* as a vertex and whose *base* is *K*.

### Lemma 2

*Suppose that*
\(m\in\mathbb{Z}^{+}\), \(\gamma,\lambda\in(0,1)\), \(\mu\in [{\lambda},{1} ]\), \(z=\mu(\bar{y},0)+(1-\mu)p\)
*for some*
\(\bar{y}\in K\), *and that*
\(\{{\bar{u}_{i}}: {i\in[m]} \} \subseteq\mathbb{R}^{n}\)
*is a set of points satisfying*

*Then there exists*
\(i\in[m]\)
*such that*

### Proof

We have

Without loss of generality we may assume that \(\bar{y}\in\bar {u}_{1}+\gamma K\). It follows that

By Lemma 1, we have

which implies that

From (1), (2), and (3) it follows that \(z\in\lambda(\bar{u}_{1},0)+(\lambda\gamma+1-\lambda)C\). □

For two numbers satisfying \(0\leq\lambda_{1}\leq\lambda_{2}\leq1\), we put

It is not difficult to verify that \(C=C_{0,1}\). And when \(0\leq\lambda_{1}\leq\lambda_{2}\leq1\) and \(\lambda_{2}\neq0\), we have

### Lemma 3

*Let*
*m*
*be a positive integer satisfying*
\(\gamma=\Gamma_{m}(K)<1\), *and*
\(0<\lambda_{1}\leq\lambda_{2}\leq1\). *Then*
\(C_{\lambda_{1},\lambda_{2}}\)
*can be covered by*
*m*
*translates of*
\((\lambda_{1}\gamma+\lambda_{2}-\lambda_{1})C\).

### Proof

If \(\lambda_{1}=\lambda_{2}\), then \(C_{\lambda_{1},\lambda_{2}}\) is a translate of \(\lambda_{2}(K\times\{0\})\) and can be covered by *m* translates of \((\lambda_{2}\gamma)C=(\lambda_{1}\gamma)C\).

Now we consider the case when \(\lambda:=\lambda_{1}\in(0,1)\) and \(\lambda_{2}=1\). There exists a set of *m* points \(\{{\bar{u}_{i}}: {i\in [m]} \}\subset\mathbb{R}^{n}\) such that

Let *z* be an arbitrary point in \(C_{\lambda,1}\). Then there exist \(\bar{y}\in K\) and \(\mu\in[\lambda,1]\) such that \(z=\mu(\bar {y},0)+(1-\mu)p\). Lemma 2 shows that there exists \(i\in[m]\) such that

It follows that \(C_{\lambda,1}\) can be covered by *m* translates of \((\lambda\gamma+1-\lambda)C\).

When \(0<\lambda_{1}<\lambda_{2}\leq1\), (4) shows that \(C_{\lambda_{1},\lambda_{2}}\) is a translate of \(\lambda_{2}(C_{\lambda_{1}/\lambda_{2},1})\), which can be covered by *m* translates of

□

### Theorem 4

*Suppose that*
\(m\in\mathbb{Z}^{+}\)
*and*
\(\gamma=\Gamma_{m}(K)\). *Then*

### Proof

Put \(\lambda=1/(2-\gamma)\). Then, by (4), \(C_{0,\lambda}\) can be covered by a translate of *λC*, and \(C_{\lambda,1}\) can be covered by *m* translates of

Thus \(\Gamma_{m+1}(C)\leq\lambda\). □

### Remark 5

When *K* is a planar convex body, we have that \(\Gamma_{7}(K)\leq1/2\). Moreover, if *K* is centrally symmetric, then we always have \(\Gamma_{7}(K)=1/2\) (see [11]). Therefore, for a convex cone *C* in \(\mathbb{R}^{3}\) having a planar convex body as a base, we have \(\Gamma_{8}(C)\leq2/3\), an estimation which was already obtained by Chuanming Zong in [9]. It is also mentioned in [11] that \(\Gamma_{7}(K)=5/11\) when *K* is a triangle. Therefore, when *T* is a tetrahedron, we have \(\Gamma _{8}(T)\leq 11/17<2/3\); see Table 1, where Δ is a triangle and *C* is a cone whose base is Δ. We have a numerical example showing that the estimation \(\Gamma_{8}(T)\leq11/17\) is not optimal either.

## 3 Covering functionals of double cones

Let \(K\subseteq\mathbb{R}^{n}\) be a convex body, *m* be a positive integer such that \(\Gamma_{m}(K)<1\), and \(p,q\in\mathbb{R}^{n+1}\setminus\mathbb {R}^{n}\times\{0\}\) be two points such that \([{p},{q} ]\cap(K\times\{0\})\neq \emptyset\). The set

is called a *double cone*.

### Theorem 6

*Let*
*C*
*be a double cone defined as above*. *Then*

### Proof

Put

Then \(\lambda\in(1/2,1)\). Without loss of generality we may assume that \([{p},{q} ]\) intersects \(K\times\{0\}\) in the origin *o* of \(\mathbb{R}^{n+1}\). By applying a suitable non-singular affine transformation if necessary, we may assume that \(p=e_{n+1}\) and \(q=-\alpha e_{n+1}\), where *α* is a positive number. Suppose that \([{\beta_{1}},{\beta_{2}} ]\subseteq [{-\alpha },{1} ]\). Put

Then it is clear that

First we show that

Let *z* be an arbitrary point in \(C_{1-\lambda}^{1}\). If \(z=p\), then *z* is clearly in \(\lambda C+(1-\lambda)p\). In the following we assume that

Then there exist \(\lambda_{1},\lambda_{2},\lambda_{3}\geq0\), and \(x\in K\times\{0\}\) such that

It follows that

On the one hand, we have

which implies that

And on the other hand, we have

which shows that (since \(o\in K\times\{0\}\))

From (5), (6), and (7) it follows that \(z\in\lambda C+(1-\lambda)p\).

In the second step we show that

Similarly, we only need to consider the case when \(-\alpha< ({z}|{e_{n+1}} )\leq-(1-\lambda)\alpha\). In this case, there exist \(\lambda_{1},\lambda_{2},\lambda_{3}\geq0\), and \(x\in K\times\{0\}\) such that

We have

Moreover, the following inequalities hold:

and

Then (8) and equalities (9), (10), and (11) show that \(z\in \lambda C+(1-\lambda)q\).

In the following we assume that \(\{{\bar{u}_{i}}: {i\in[m]} \}\subseteq\mathbb{R}^{n}\) is a set of points such that

Suppose that \(({z}|{e_{n+1}} )\in [{0},{1-\lambda } ]\). Then there exist \(\lambda_{1},\lambda_{2},\lambda_{3}\geq0\), \(x\in K\times\{0\}\) such that

In this situation we have

Moreover,

which shows that

Put

Then \(\mu\in [{\lambda},{1} ]\). From Lemma 2 it follows that \(z\in\lambda(\bar{u}_{i},0)+\lambda C\) for some \(i\in[m]\). Therefore

It remains to consider the case when \(({z}|{e_{n+1}} )\in [{-(1-\lambda)\alpha },{0} ]\). Then there exist \(\lambda_{1},\lambda_{2},\lambda_{3}\geq0\), \(x\in K\times\{0\}\) such that

We have

Clearly,

which shows that

Moreover, one can easily verify that

Again, by Lemma 2, \(z\in\lambda(\bar {u}_{i},0)+\lambda C\) for some \(i\in[m]\). Therefore

□

In Table 2, *C* is a double cone whose base is \(K_{1}^{2}\) (by \(K_{p}^{n}\) we denote the unit ball of the Banach space \(l_{p}^{n}\)). Compared with exact values of \(\Gamma_{m}(K_{1}^{3})\), the estimations of \(\Gamma_{m}(C)\) given by Theorem 6 are optimal for \(m=6,7,8\).

## 4 Conclusion

Let *K* be a convex body in \(\mathbb{R}^{n}\) and *C* be a compact convex cone in \(\mathbb{R}^{n+1}\) having \(K\times\{0\}\) as a base. We proved that

A similar estimation is also provided for double cones. These estimations are optimal for particular pairs of *m* and *K*, are better than existing estimations, but they are not always optimal.

In the authors’ opinion, it is interesting to do the following: provide better estimations of the covering functionals of cones and double cones, characterize convex bodies that are sufficiently close to cones or double cones via their boundary structure, and, more importantly, get precise values of \(\Gamma_{2^{n}}(K)\) when *K* is an *n*-simplex or \(K_{1}^{n}\) for \(n\geq3\).

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## Acknowledgements

The authors are grateful to Chan He for her useful comments and suggestions, and we would like to thank one of the reviewers for reminding us of the connection between the covering functional and the entropy number.

## Funding

Senlin Wu is supported by the National Natural Science Foundation of China, grant numbers 11371114 and 11571085.

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Wu, S., Xu, K. Covering functionals of cones and double cones.
*J Inequal Appl* **2018**, 186 (2018). https://doi.org/10.1186/s13660-018-1785-9

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DOI: https://doi.org/10.1186/s13660-018-1785-9