Abstract
We introduce a new geometric constant \(C_{NJ}^{(p)}(X)\) for a Banach space X, called a generalized von Neumann-Jordan constant. Next, it is shown that \(1\leq C_{NJ}^{(p)}(X)\leq2\) for any Banach space X and that the right hand side inequality is sharp if and only if X is uniformly non-square. Moreover, a relationship between the James constant \(J(X)\) and \(C_{NJ}^{(p)}(X)\) is presented. Finally, the generalized von Neumann-Jordan constant of the Lebesgue space \(L_{r}([0,1])\) is calculated and a relationship between \(C_{NJ}^{(p)}(X)\) and the fixed point property is found.
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1 Introduction
Recently many geometric constants for a Banach space X have been investigated. In particular, the von Neumann-Jordan constant \(C_{NJ}(X)\) and the James constant \(J(X)\) are widely treated. We introduce a new geometric constant, called the generalized von Neumann-Jordan constant \(C_{NJ}^{(p)}(X)\), which is related to the von Neumann-Jordan constant of a Banach space X and can be used for much better characterization of a Banach space X.
In connection with the famous work [1] (see also [2]) of Jordan and von Neumann concerning inner products, the von Neumann-Jordan constant \(C_{NJ}(X)\) for a Banach space X was introduced by Clarkson [3] as the smallest constant C, for which the estimates
hold for all \(x,y\in X \) with \((x,y)\neq(0,0)\). Equivalently,
The classical von Neumann-Jordan constant \(C_{NJ}(X)\) was investigated in many papers (see for instance [4–7]).
A Banach space X is said to be uniformly non-square in the sense of James if there exists a positive number \(\delta<2\) such that for any \(x,y\in S_{X}:= \{ x\in X\colon\|x\| = 1 \}\), we have
The James constant \(J(X)\) of a Banach space X is defined by
It is obvious that X is uniformly non-square if and only if \(J(X)<2\).
In this paper we introduce a new constant \(C_{NJ}^{(p)}(X)\), generalizing the von Neumann-Jordan constant \(C_{NJ}(X)\). By the definition of \(C_{NJ}^{(p)}(X)\), we will get a relationship between \(C_{NJ}^{(p)}(X)\) and \(J(X)\), as well as we will estimate the value of \(C_{NJ}^{(p)}(X)\). Furthermore, the constant \(C_{NJ}^{(p)}(X)\) enable us to establish some new equivalent conditions for the uniform non-squareness of a Banach space X. Since any uniformly non-square Banach space X has the fixed point property (see [8]), our constant \(C_{NJ}^{(p)}(X)\) is related to the fixed point theory. Moreover, the value of the generalized von Neumann-Jordan constant for the space \(L_{r}[0,1]\) will be calculated. Finally, we will find a relationship between the constant \(C_{NJ}^{(p)}(X)\) and normal structure of X, and in such a way we have again its relationship to the fixed point theory.
2 Preliminaries
Let \(X = (X, \|\cdot\|)\) be a real Banach space. Geometrical properties of a Banach space X are determined by its unit sphere \(S_{X}\) or its unit ball \(B(X)\).
Definition 1
The generalized von Neumann-Jordan constant \(C_{NJ}^{(p)}(X)\) is defined by
where \(1\leq p<\infty\).
We will also use the following parametrized formula for the constant \(C_{NJ}^{(p)}(X)\) (see [9] and [7] in the case of the classical von Neumann-Jordan constant):
where \(1\leq p<\infty\). By taking \(t=1\) and \(x=y\), we obtain the estimate
Definition 2
(see [10])
The modulus of uniform smoothness of X is defined as
It is clear that \(\rho_{X}(t)\) is a convex function on the interval \([0,\infty)\) satisfying \(\rho_{X}(0)=0\), whence it follows that \(\rho_{X}\) is nondecreasing on \([0,\infty)\). It is also easy to show that \(\max \{0,t-1 \}\leq\rho_{X}(t)\leq t\).
Definition 3
(see [11])
A Banach space X is said to be uniformly smooth if \((\rho_{X})'_{+}(0):=\lim_{t \to0^{+}}\frac{\rho_{X}(t)}{t}=0\).
Definition 4
A Banach space X is said to be q-uniformly smooth (\(1< q\leq2\)) if there exists a constant \(K>0\) such that \(\rho_{X}(t)\leq K t^{q}\) for all \(t>0\).
Definition 5
(see [13])
Given any Banach space X and a number \(p\in[1,\infty)\), another function \(J_{X,p}(t)\) is defined by
on the interval \([0,\infty)\).
By the inequality
which follows by convexity of the function \(f(u)=u^{p}\) on \([0,\infty)\), we get \(J_{X,p}(t)\geq\rho_{X}(t)+1\) when \(1\leq p<\infty\). For \(p=1\) and \(p=2\), we have the equalities \(J_{X,1}(t)=\rho_{X}(t)+1\) and \(2J_{X,2}^{2}(t)=E(t,X)\), respectively, where the constant \(E(t,X)\) was introduced by Gao [14] in 2005, and it is defined by the formula
Definition 6
(see [15])
For any Banach space X, we define
Definition 7
A Banach space X is said to have normal (resp. weak normal) structure if X contains no bounded and closed (resp. weakly compact) convex subset C with more than one point which is diametral in the sense that, for all \(x\in C\),
Recall that the weak normal structure (so the normal structure as well) of a Banach space X implies the weak fixed point property for X (see [16, 17]).
Remark 2.1
(see [18])
A sufficient condition for normal structure of a Banach space X is the following: there exists \(\varepsilon\in(0,2)\) such that
where \(\delta_{x}:[0,2]\rightarrow[0,1]\) is the classical modulus of convexity of X defined as
Lemma 2.2
(see [13])
For any Banach space X and any \(1\leq p<\infty\) the following statements are true:
-
(1)
\(J_{X,p}(\cdot)\) is nondecreasing on \((0,\infty)\).
-
(2)
\(J_{X,p}(\cdot)\) is convex on \((0,\infty)\).
-
(3)
\(J_{X,p}(\cdot)\) is continuous on \((0,\infty)\).
-
(4)
\(\frac{J_{X,p}(\cdot)-1}{t}\) is nondecreasing on \((0,\infty)\).
The proof of this lemma can be found in [13].
Lemma 2.3
For any \(1\leq p<\infty\) a Banach space X is uniformly smooth if and only if \(\lim_{t \to0^{+}}\frac{J_{X,p}(t)-1}{t}=0\).
Proof
Since \(J_{X,p}(t)\geq\rho_{X}(t)+1\) for any \(t>0\) and \(1\leq p<\infty\), the sufficiency is obvious. Now we will prove the necessity. Assume, to derive a contradiction, that \(\lim_{t \to0^{+}}\frac {J_{X,p}(t)-1}{t}>0\). By Lemma 2.2(4), there exists \(0< c<1\) such that \(\lim_{t \to0^{+}}\frac {J_{X,p}(t)-1}{t}\geq c\). In particular, we can choose \(0< t<1\) and x, y in X with \(\|x\|=1\), \(\|y\|=t\) satisfying
We can assume without loss of generality that \(\min\{\|x+y\|,\|x-y\|\} =\|x-y\|\). Then, denoting \(\|x-y\|=h\), we have \(h\in[1-t,1+t]\), which follows from the inequalities \(\vert \|x\|-\|y\|\vert \leq\|x-y\|\leq\|x\|+\|y\|\). By inequality (2.1), we obtain
Since
it is easy to see that f is an increasing function with respect to h on the interval \([1-t,1+ct]\) and decreasing on the interval \([1+ct,1+t]\). Hence the minimum value of the function \(f(h)\) can be attained either at \(h=1-t\) or at \(h=1+t\). In the case when the minimum value is attained at the point \(1-t\), we have by the definition of the modulus of uniform smoothness that
In the second case, we have
In both cases, letting \(t\rightarrow0^{+}\) and using the L’Hôpital rule, we easily obtain \(\lim _{t \to0^{+}}\frac{\rho _{X}(t)}{t}\geq c>0\). Obviously, this contradicts the definition of uniform smoothness of X, and thus we completed the proof. □
Lemma 2.4
(see [12])
Let \(1\leq p< \infty\) and \(1< q \leq2\). A Banach space X is q-uniformly smooth if and only if there exists a constant \(K\geq1\) such that
Therefore, according to Lemma 2.4 and the definition of \(J_{X,p}(\cdot)\), the following lemma holds.
Lemma 2.5
Let \(1\leq p< \infty\) and \(1< q \leq2\). The following statements are equivalent:
-
(1)
X is q-uniformly smooth.
-
(2)
There exists a constant \(K\geq1\) such that the inequality \(J_{X,p}(t)\leq(1+Kt^{q})^{\frac{1}{q}}\) is satisfied for any \(t>0\).
3 Main results
Theorem 3.1
For any Banach space X and any \(1\leq p<\infty\) the generalized von Neumann-Jordan constant \(C_{NJ}^{(p)}(X)\) satisfies the inequality \(C_{NJ}^{(p)}(X)\leq2\).
Proof
We will use in the proof the following parametrized formula for the generalized von Neumann-Jordan constant \(C_{NJ}^{(p)}(X)\), where \(1\leq p<\infty\):
Since
so
Applying convexity of the function \(\varphi(u)=|u|^{p}\), we get
Combining this estimate with inequality (3.1), we get
Hence
and the proof is completed. □
Lemma 3.2
(see [6])
Let \(1< p<\infty\). A Banach space X is uniformly non-square if and only if there exists \(\delta\in(0,1)\) such that for any \(x,y\in X\), we have
According to Lemma 3.2, we directly obtain the following theorem.
Theorem 3.3
Let \(1\leq p<\infty\). A Banach space X is uniformly non-square if and only if \(C_{NJ}^{(p)}(X)<2\).
Now let us present the following theorem indicating the relationship between constants \(J(X)\) and \(C_{NJ}^{(p)}(X)\).
Theorem 3.4
For any \(1< p<\infty\) and any Banach space X, the following inequality holds:
Proof
Indeed, if \(1< p<\infty\), then for any \(x,y\in S_{X}\), we have
so
and the proof is completed. □
By Theorem 3.4, we obtain the following corollary.
Corollary 3.5
For any Banach space X and any \(1\leq p<\infty\) the inequalities \(C_{NJ}^{(p)}(X)<2\) and \(J(X)<2\) are equivalent. Moreover, if X is a Banach space with \(C_{NJ}^{(p)}(X)<2\), then X has the fixed point property.
Proof
It is well known that \(J(X)<2\) if and only if a Banach space X is uniformly non-square. However, by Theorem 3.3, we know that a Banach space X is uniformly non-square if and only if \(C_{NJ}^{(p)}(X)<2\). Hence, \(J(X)<2\) if and only if \(C_{NJ}^{(p)}(X)<2\). Moreover, every uniformly non-square Banach space have the fixed point property (see [8]), so if X is a Banach space with \(C_{NJ}^{(p)}(X)<2\), then X has the fixed point property. □
Now we will calculate the generalized von Neumann-Jordan constant for the space \(L_{r}[0,1]\).
Theorem 3.6
Let X be the Banach space \(L_{r}[0,1]\). Let \(1< r\leq2\) and \(\frac {1}{r}+\frac{1}{r^{\prime}}=1\). Then
-
(1)
if \(1< p\leq r\) then \(C_{NJ}^{(p)}(L_{r}[0,1])=2^{2-p}\) and if \(r< p\leq r^{\prime}\) then \(C_{NJ}^{(p)}(L_{r}[0,1])=2^{\frac{p}{r}-p+1}\);
-
(2)
if \(r^{\prime}< p<\infty\) then \(C_{NJ}^{(p)}(L_{r}[0,1])=1\).
Proof
Let us note that \(r\leq2\leq r'\) and
(1) for any \(x,y\in S_{X}\) and any \(0\leq t\leq1\), if \(1< p\leq r'\), then in virtue of Remark 2.3 from [19], we have
which is equivalent to
Consequently,
whence
and from the definition of \(C_{NJ}^{(p)}(L_{r}[0,1])\), we have
Defining \(f(t)=\frac{(1+t^{r})^{\frac{p}{r}}}{1+t^{p}}\), we get \((f(t))^{r}=\frac{(1+t^{r})^{p}}{(1+t^{p})^{r}}=:G(t)\). Obviously, both functions \(f(t)\) and \(G(t)\) are continuous and
whence it follows that \(G'(t)=0\) if and only if
i.e. \(t^{r}(1+t^{p})-t^{p}(1+t^{r})=0\), which means that \(t^{r}=t^{p}\). Let us observe that if \(p=r\), then \(G(t)=1\) for any \(t\in[0,1]\), so \(G'(t)=0\) on the whole interval \([0,1]\).
Notice also that if \(1< p\neq r\), then there is no interior point of the interval \([0,1]\) at which the derivative \(G'(t)\) vanishes. Therefore, the function \(f(t)\) can reach its biggest value on the interval \([0,1]\) either at the point 0 (\(f(0)=1\)) or at the point 1 (\(f(1)=2^{\frac {p}{r}-1}\)), depending on the relationship between p and r. Namely:
-
if \(1< p\leq r\), then \(2^{\frac{p}{r}-1}\leq1\), so \(C_{NJ}^{(p)}(L_{r}[0,1])\leq\frac{2}{2^{p-1}}\cdot1=2^{2-p} \);
-
if \(r< p\leq r'\), then \(2^{\frac{p}{r}-1}>1\), so \(C_{NJ}^{(p)}(L_{r}[0,1])\leq\frac{2}{2^{p-1}}\cdot2^{\frac {p}{r}-1}=2^{\frac{p}{r}-p+1}\).
On the other hand, notice that the space \(L_{r}[0,1]\) is r-uniformly smooth if \(1< r\leq2\), and the following Clarkson inequality is satisfied:
If \(1< p\leq r'\), the thesis in Lemma 2.4 holds with \(K=1\). Therefore, we have the inequality \(J_{X,p}(t)\leq(1+t^{r})^{\frac{1}{r}}\) for any \(t\geq0\). Take x and y from the space \(L_{r}[0,1]\), satisfying \(\int_{0}^{b}|x(s)|^{r} \,ds=1\) and \(\int_{b}^{1}|y(s)|^{r} \,ds=1\) with some \(b\in(0,1)\) and let
Then \(\|x_{1}(s)\|_{r}=\|y_{1}(s)\|_{r}=1\), and if \(1< p<r'\), we have
Thus
which means that if \(1< p\leq r'\). Therefore
Taking \(t=1\), we get \(C_{NJ}^{(p)}(L_{r}[0,1])\geq2^{\frac{p}{r}-p+1} \), while taking \(t=0\), we obtain \(C_{NJ}^{(p)}(L_{r}[0,1])\geq2^{2-p} \). Therefore:
-
if \(1< p\leq r\) then \(2^{2-p}\geq2^{\frac{p}{r}-p+1}\) and \(C_{NJ}^{(p)}(L_{r}[0,1])\geq2^{2-p}\);
-
if \(r< p\leq r'\) then \(2^{\frac{p}{r}-p+1}> 2^{2-p}\) and \(C_{NJ}^{(p)}(L_{r}[0,1])\geq2^{\frac{p}{r}-p+1}\).
From what has been discussed above, the results from the thesis (1) of the theorem follow immediately.
(2) In the case when \(r'< p<\infty\), in virtue of Remark 2.3 from [19] we know that for any \(x,y\in S_{X}\) and any \(0\leq t\leq1\), we have
which is equivalent to
Consequently,
By the proof of thesis (1), if \(r< p\) then the supremum of the function f is equal to \(2^{\frac{p}{r}-1}\), so we have
By the observation just after Definition 1 of \(C_{NJ}^{(p)}(X)\), we have \(C_{NJ}^{(p)}(X)\geq1\), so thesis (2) is proved and the proof of the theorem is completed. □
The following theorem gives a relationship between the constant \(C_{NJ}^{(p)}(X)\) and the normal structure of X. It is a generalization of a similar result from [20] concerning only the case \(p=2\).
Theorem 3.7
If \(1\leq p<\infty\) and X is a Banach space with \(C_{NJ}^{(p)}(X)<\frac{1}{2^{p-1}}(1+\frac{1}{\mu(X)})^{p}\), then X has normal structure.
Proof
Let us observe that by the inequality \(\mu(X)\geq1\), we have \(C_{NJ}^{(p)}(X)<2\). We know that if \(J(X)<2\), then X is reflexive (see [21]). Therefore, by Corollary 3.5, \(C_{NJ}^{(p)}(X)<2\), and so X is reflexive and it has normal structure if and only if it has weak normal structure.
Looking for a contradiction, suppose that X fails to have weak normal structure. Then it is well known (see [17]) that there exists a bounded sequence \((x_{n})\) in X satisfying the following statements:
-
(i)
\((x_{n})\) is weakly convergent to 0 in X,
-
(ii)
\(\operatorname{diam}(\{x_{n}:n=1,2,\ldots\})=1\),
-
(iii)
for all \(x\in\overline{\operatorname{conv}} (\{x_{n}:n=1,2,\ldots\})\), we have
$$\begin{aligned} \lim_{n\rightarrow\infty}\| x-x_{n}\|=\operatorname {diam} \bigl(\{x_{n}:n=1,2,\ldots\}\bigr)=1. \end{aligned}$$
Let us fix \(\varepsilon>0\) as small as needed. Then, using the above properties of \((x_{n})\) and the definition of \(\mu:=\mu(X)\), we can find two positive integers n, m, with \(m>n\), such that
-
(1)
\(\| x_{n}\|\geq1-\varepsilon\),
-
(2)
\(\| x_{m}-x_{n}\|\leq1\),
-
(3)
\(\|x_{m}+x_{n}\|\leq\mu+\varepsilon\),
-
(4)
\(\|(1+\frac{1}{\mu+\varepsilon})x_{m}-(1-\frac{1}{\mu +\varepsilon})x_{n}\|\geq(1+\frac{1}{\mu+\varepsilon})(1-\varepsilon)\),
-
(5)
\(\|(1-\frac{1}{\mu+\varepsilon})x_{m}-(1+\frac{1}{\mu +\varepsilon})x_{n}\|\geq(1+\frac{1}{\mu+\varepsilon})\|x_{n}\|-\varepsilon\).
Since
by condition (2), when m is big enough, we get
and condition (3) is proved. We just need to prove conditions (4) and (5).
Let us fix \(n\in\mathbb{N}\) and define again \(\mu:=\mu(X)\). Notice that we can easily get from the Mazur theorem
for any \(n\in\mathbb{N}\). Indeed, since \(x_{n}\rightarrow0\) weakly as \(n\rightarrow\infty\), then by the Mazur theorem \(0\in\overline{\operatorname {conv}}(\{x_{k}:k\in\mathbb{N}\})\), whence (3.2) follows immediately. Since (3.2) holds, so by the assumption that X fails to have weak normal structure, for some \(m>n\), we have
and condition (4) follows. In the same way, we can get condition (5).
Next, put \(x=x_{m}-x_{n}\), \(y=(\mu+\varepsilon)^{-1}(x_{m}+x_{n})\) and use the previous estimates to obtain \(\|x\|\leq1\), \(\|y\|\leq1\), and
By the definition of \(C_{NJ}^{(p)}(X)\), we get the estimate
Finally, letting \(\varepsilon\rightarrow0^{+}\), we obtain
which contradicts the hypothesis. This contradiction finishes the proof of the theorem. □
References
Jordan, P, von Neumann, J: On inner products in linear metric spaces. Ann. Math. 36, 719-723 (1935)
Zuo, Z, Cui, Y: Some sufficient conditions for fixed points of multivalued nonexpansive mappings. Fixed Point Theory Appl. 2009, Article ID 319804 (2009). doi:10.1155/2009/319804
Clarkson, JA: The von Neumann-Jordan constant for the Lebesgue space. Ann. Math. 38, 114-115 (1937)
Kato, M, Maligranda, L, Takahashi, Y: On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces. Stud. Math. 144(3), 275-295 (2001)
Kato, M, Maligranda, L: On James and Jordan-von Neumann constants of Lorentz sequence spaces. J. Math. Anal. Appl. 258(2), 457-465 (2001)
Takahashi, Y, Kato, M: Von Neumann-Jordan constant and uniformly non-square Banach spaces. Nihonkai Math. J. 9, 155-169 (1998)
Yang, C: Jordan-von Neumann constant for Banaś-Frączek space. Banach J. Math. Anal. 8(2), 185-192 (2014)
García-Falset, J, Llorens-Fuster, E, Mazcuńan-Navarro, EM: Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233, 494-514 (2006)
Yang, C, Wang, F: On a new geometric constant related to the von Neumann-Jordan constant. J. Math. Anal. Appl. 324(1), 555-565 (2006)
Lindenstrauss, J: On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J. 10, 241-252 (1963)
Day, MM: Uniform convexity in factor and conjugate spaces. Ann. Math. 45, 375-385 (1944)
Takahashi, Y, Hashimoto, K, Kato, M: On sharp uniform convexity, smoothness and strong type, cotype inequalities. J. Nonlinear Convex Anal. 3(2), 267-281 (2002)
Zuo, Z, Cui, Y: A coefficient related to some geometric properties of a Banach space. J. Inequal. Appl. 2009, Article ID 934321 (2009). doi:10.1155/2009/934321
Gao, J: Normal structure and Pythagorean approach in Banach space. Period. Math. Hung. 51(2), 19-30 (2005)
Jiménez-Melado, A, Llorens-Fuster, E: The fixed point property for some uniformly nonsquare Banach spaces. Boll. Unione Mat. Ital., A (7) 10, 587-595 (1996)
Kirk, WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72, 1004-1006 (1965)
Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Kirk, WA, Sims, B: Handbook of Metric Fixed Point Theory. Kluwer Academic, Dordrecht (2001)
Kato, M, Takahashi, Y: Type, cotype constants and Clarkson’s inequalities for Banach spaces. Math. Nachr. 186, 187-195 (1997)
Jiménez-Melado, A, Llorens-Fuster, E, Saejung, S: The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc. Am. Math. Soc. 134(2), 355-364 (2005)
James, RC: Uniformly non-square Banach spaces. Ann. Math. 80, 542-550 (1964)
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The first author gratefully acknowledges the support of the NFSC (No. 11401143).
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Cui, Y., Huang, W., Hudzik, H. et al. Generalized von Neumann-Jordan constant and its relationship to the fixed point property. Fixed Point Theory Appl 2015, 40 (2015). https://doi.org/10.1186/s13663-015-0288-3
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DOI: https://doi.org/10.1186/s13663-015-0288-3