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Covering functionals of cones and double cones
- Senlin Wu^{1}Email authorView ORCID ID profile and
- Ke Xu^{1}
https://doi.org/10.1186/s13660-018-1785-9
© The Author(s) 2018
- Received: 19 April 2018
- Accepted: 18 July 2018
- Published: 24 July 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.
Keywords
- Banach–Mazur distance
- Compact convex cones
- Convex body
- Covering
- Hadwiger’s covering conjecture
MSC
- 52A20
- 52A15
- 52A40
- 52C17
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 intK and bdK, 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 intK 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.
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
Proof
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
Proof
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 \((\lambda_{2}\gamma)C=(\lambda_{1}\gamma)C\).
Theorem 4
Proof
Remark 5
m | 3 | 4 | 5 | 6 | 7 | 8 |
\(\Gamma_{m}(\Delta)\) | \(\frac{2}{3}\) | \(\frac{4}{7}\) | \(\frac {8}{15}\) | \(\frac{1}{2}\) | \(\frac{5}{11}\) | \(\frac{3}{7}\) |
\(\Gamma_{m}(T)\) | 1 | \(\frac{3}{4}\) | \(\frac{9}{13}\) | – | – | – |
\(\Gamma_{m}(C)\) | 1 | \({\leq}\frac{3}{4}\) | \({\leq}\frac{7}{10}\) | \({\leq}\frac{15}{22}\) | \({\leq}\frac{2}{3}\) | \({\leq}\frac{11}{17}\) |
3 Covering functionals of double cones
Theorem 6
Proof
m | 3 | 4 | 5 | 6 | 7 | 8 |
\(\Gamma_{m}(K_{1}^{2})\) | 1 | \(\frac{1}{2}\) | \(\frac{1}{2}\) | \(\frac {1}{2}\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) |
\(\Gamma_{m}(K_{1}^{3})\) | 1 | 1 | 1 | \(\frac{2}{3}\) | \(\frac{2}{3}\) | \(\frac{2}{3}\) |
\(\Gamma_{m}(C)\) | 1 | 1 | 1 | \(\leq\frac{2}{3}\) | \(\leq\frac{2}{3}\) | \(\leq\frac{2}{3}\) |
4 Conclusion
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 \(n\geq3\).
Declarations
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.
Authors’ contributions
Both the authors contributed equally to this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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