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A new upper bound of geometric constant \(D(X)\)
- Jin Huan Li^{1}Email authorView ORCID ID profile,
- Bo Ling^{1} and
- San Yang Liu^{1}
https://doi.org/10.1186/s13660-017-1474-0
© The Author(s) 2017
- Received: 25 March 2017
- Accepted: 6 August 2017
- Published: 31 August 2017
Abstract
A new constant \(\mathit{WD}(X)\) is introduced into any real \(2^{n}\)-dimensional symmetric normed space X. By virtue of this constant, an upper bound of the geometric constant \(D(X)\), which is used to measure the difference between Birkhoff orthogonality and isosceles orthogonality, is obtained and further extended to an arbitrary m-dimensional symmetric normed linear space (\(m\geq2\)). As an application, the result is used to prove a special case for the reverse Hölder inequality.
Keywords
- Birkhoff orthogonality
- isosceles orthogonality
- symmetric normed linear spaces
- geometric constant
1 Introduction
The notion of orthogonality has many forms when the underlying space is transferred from inner product spaces to real normed spaces. For example, Birkhoff [1] introduced Birkhoff orthogonality in which X is assumed to be a real normed linear space. If \(\Vert x+\lambda y \Vert \geq \Vert x \Vert \), \(\forall\lambda\in \mathbb{R}\), then x is said to be Birkhoff orthogonal to y. It can be written as \(x \perp_{B} y\). James [2] defined isosceles orthogonality, that is, if \(\Vert x+y \Vert =\Vert x-y \Vert \), then x is said to be isosceles orthogonal to y. It is denoted by \(x\perp_{I} y\). When X is an inner product space, these two types of orthogonality are equivalent to inner-product orthogonality.
In this study, by considering the constant \(D(X)\) in \(2^{n}\)-dimensional real symmetric normed linear spaces, we obtain an upper bound \(\mathit{WD}(X)\). As we discuss in Corollary 1, this bound can be extended to any m-dimensional symmetric normed linear space (\(m\geq 2\)). This article is organized as follows. In Section 2, we present some notations and definitions. In Section 3, the constant \(\mathit{WD}(X)\) is introduced and discussed. In Section 4, we consider \(\mathit{WD}(X)\) for the space \(l_{p}^{2^{n}}\) and present a special case for the reverse Hölder inequality.
2 Preliminaries
Let us fix some notations. Let X be an n-dimensional real linear normed space. By \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert ^{\ast}\), we denote the norm of X and the norm of a dual space \(X^{\ast}\), respectively. The notation \(S(X)\) is the unit sphere of X. Let \(\mathbb{R}\) and \(\mathbb{N}\) denote the real field and a positive integer set, respectively.
Definition
Let X be an n-dimensional symmetric normed linear space and \(e_{1}, \ldots, e_{n}\) be a group of symmetric axes. For \(x\in X\), x is denoted by the coordinate representation of this group of symmetric axes, i.e., \(x=(x_{1},\ldots,x_{n})=x_{1}e_{1}+\cdots+x_{n}e_{n}\).
3 Main results
Firstly, the following elementary results are presented. Throughout this paper, the symbol \(\langle\cdot, \cdot\rangle\) denotes the natural inner product of two n-dimensional vectors. The first two lemmas are known, but we fail to find literature sources.
Lemma 1
Let X be a normed space \((\mathbb{R}^{n}, \Vert \cdot \Vert )\) and \(e_{1}=(1,0,\ldots, 0)\), \(e_{2}=(0,1,0,\ldots, 0), \ldots, e_{n}=(0,0,\ldots, 0,1)\) be a basis of X. Assume that B is a skew-symmetric matrix, i.e., \(B^{T}=-B\). Then \(\langle x,Bx\rangle=0\) for any \(x\in X\).
Proof
Given that \(\langle x,Bx\rangle=\langle B^{T}x,x\rangle=-\langle x,Bx\rangle\), then \(\langle x,Bx\rangle=0\). □
Lemma 2
- (i)
each square matrix \(B_{2^{n},i}\) (\(i=1, 2, \ldots,2^{n}-1\)) is a skew-symmetric orthogonal matrix, i.e., \(B_{2^{n},i}^{T}=-B_{2^{n},i}\), \(B_{2^{n},i}^{T}B_{2^{n},i}=Id_{2^{n}}\);
- (ii)
each row and column in \(B_{2^{n},i}\) (\(i=1, 2, \ldots, 2^{n}-1\)) has one and only one non-zero element, and this element is 1 or −1;
- (iii)
matrices \(B_{2^{n},i}^{T} B_{2^{n},j}\), \(i\neq j\), i, \(j=1, 2, \ldots, 2^{n}-1\), satisfy the preceding two properties, i.e., (i) and (ii).
Proof
Assume that this lemma holds for \(n=k\), namely, the matrices \(B_{2^{k},1}\), \(B_{2^{k},2}, \dots, B_{2^{k},2^{k}-1}\) exist, satisfying the conditions (i) to (iii).
In order to present an upper bound of \(D(X)\), a new constant \(\mathit{WD}(X)\) for any real normed linear space \(X=(\mathbb{R}^{2^{n}}, \Vert \cdot \Vert )\) is introduced.
Definition 1
Proposition 1
Proof
Lemma 3
- (1)
Assume that each row and each column in the \(2^{n}\times 2^{n}\) matrix B has only one non-zero element, which takes the value of 1 or −1. Then matrix B is an isometric operator on X.
- (2)Assume that
- (i)
B is a skew-symmetric and orthogonal matrix, i.e., \(B^{T}=-B\), \(B^{T}B=Id\);
- (ii)
Each row and each column in matrix B has only one non-zero element, which takes the value of 1 or −1.
- (i)
Proof
(1) The equality \(y=Bx\) indicates that y is merely the vector in which the elements are a rearrangement of the corresponding elements of x; some items in the elements change their sign. Thus, based on the definition of a real symmetric linear normed space, we have \(\Vert Bx \Vert =\Vert x \Vert \).
(2) Let \(y=Bx\), by Lemma 3(1), B is an isometric operator. Thus, \(\Vert Bx \Vert =\Vert x \Vert =1\). Meanwhile, \(\Vert x+y \Vert =\Vert (Id+B)x \Vert =\Vert B(Id-B)x \Vert =\Vert (Id-B)x \Vert =\Vert x-y \Vert \), and \(x \perp_{I} y\) are obtained. □
Then the main theorem can be obtained.
Theorem 1
Let X be a real symmetric linear normed space \((\mathbb{R}^{2^{n}}, \Vert \cdot \Vert )\) and \(e_{1}=(1,0,\ldots, 0)\), \(e_{2}=(0,1,0,\ldots, 0), \dots, e_{2^{n}}=(0,0,\ldots, 0,1)\) be a basis of X. And the normed space X is such that, for any \(x\in S(X)\) and any \(y=B_{2^{n},i}x\), \(B_{2^{n},i}\in\{B_{2^{n},1}, \ldots, B_{2^{n},2^{n}-1}\}\), there exists \(\lambda_{0}\in\mathbb{R}\) such that \(x+\lambda_{0} y \perp_{B} H\), where \(H=\operatorname{span}\{y, B_{2^{n},1}x, \ldots, B_{2^{n},i-1}x, B_{2^{n},i+1}x,\ldots, B_{2^{n},2^{n}-1}x\}\). Then \(2(\sqrt{2}-1 )\leq D(X) \leq \mathit{WD}(X)\leq1\).
Proof
The first inequality has been proven in Theorem 1 of [3]; thus, the last inequality can be easily obtained. The second inequality can be proven as follows by assuming that \(x\in S(X)\). Given that X is a symmetric normed linear space and \(B_{2^{n},i}\), \(i=1,\ldots, 2^{n}-1\), satisfies properties (i) and (ii) in Lemma 2. By Lemma 3(2), \(B_{2^{n},i} x \in S(X)\) and \(x\perp _{I} B_{2^{n},i}x\) (\(i=1, \ldots, 2^{n}-1\)) can be obtained. Hence, we get \(D(X) \leq \mathit{WD}(X)\). □
It is easy to extend the above result to any m-dimensional real symmetric normed linear space.
Corollary 1
Let X be a real symmetric linear normed space \((\mathbb{R}^{m}, \Vert \cdot \Vert )\) and \(e_{1}=(1,0,\ldots, 0)\), \(e_{2}=(0,1,0,\ldots, 0), \dots, e_{m}=(0,0,\ldots, 0,1)\) be a basis of X. And the normed space X is such that there exists a subspace \(Y\subset X\) with \(\operatorname{dim} Y=2^{n}\) and, for any \(x\in S(Y)\) and any \(y=B_{2^{n},i}x\), \(B_{2^{n},i}\in\{B_{2^{n},1}, \ldots, B_{2^{n},2^{n}-1}\}\), there exists \(\lambda_{0}\in\mathbb{R}\) such that \(x+\lambda_{0} y \perp_{B} H\), where \(H=\operatorname{span}\{y, B_{2^{n},1}x, \ldots, B_{2^{n},i-1}x, B_{2^{n},i+1}x,\ldots, B_{2^{n},2^{n}-1}x\}\). Then \(2(\sqrt{2}-1 )\leq D(X) \leq \mathit{WD}(Y)\leq1\).
It is worth mentioning that the upper bound \(\mathit{WD}(X)\) of the geometric constant \(D(X)\), which is given in Theorem 1, has several advantages. Firstly, it is defined unrelated to isosceles orthogonality compared to \(D(X)\). Secondly, due to (2), \(\mathit{WD}(X)\) has a simple expression, which makes calculation feasible. Finally, it is less than one in general. For example, we consider \(\mathit{WD}(X)\) for the space \(l_{p}^{2^{n}}\) in the next section.
4 The case of \(l_{p}^{2^{n}}\)
The space \(l_{p}^{2^{n}}\) is used to show that the aforementioned upper bound \(\mathit{WD}(X)\) is optimal for \(D(X)\).
Proposition 2
Proof
Remark 1
According to the above proof, \(\mathit{WD}(l_{p}^{2^{n}})\) is independent of the selection of B. Thus, it may verify the existence of the space X satisfying the condition of Proposition 1.
Corollary 2
Corollary 3
\(\lim_{p\rightarrow\infty}D(l_{p}^{m})=2(\sqrt{2}-1 )\). Specially, \(\mathit{WD}(l_{1}^{2^{n}})=2(\sqrt{2}-1 )\).
Corollary 4
5 Conclusion
In this paper, by studying the geometric constant \(D(X)\) of any real \(2^{n}\)-dimensional symmetric normed space \(X=(\mathbb{R}^{2^{n}}, \Vert \cdot \Vert )\), we obtained an upper bound \(\mathit{WD}(X)\), which is not greater than 1. And using the special properties of a finite dimensional normed space \((\mathbb{R}^{2^{n}}, \Vert \cdot \Vert )\) and the constraints on \((\mathbb{R}^{2^{n}}, \Vert \cdot \Vert )\), we also give a simple formula for \(\mathit{WD}(X)\). In particular, when \(X = l_{p}^{2^{n}}\), this formula is used to give a special form of the reverse Hölder inequality.
Declarations
Acknowledgements
The second author is supported by the National Natural Science Foundation of China (Grant No. 11401451). The third author is supported by the National Natural Science Foundation of China (Grant No. 61373174).
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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