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On some inequalities in 2-metric spaces
Journal of Inequalities and Applications volume 2021, Article number: 130 (2021)
Abstract
In this paper, we establish new inequalities in the setting of 2-metric spaces and provide their geometric interpretations. Some of our results are extensions of those obtained by Dragomir and Goşa (J. Indones. Math. Soc. 11(1):33–38, 2005) in the setting of metric spaces.
1 Introduction and preliminaries
We start this section by recalling an interesting metric-type inequality due to Dragomir and Goşa [7]. Let us first fix some notations. We denote by \(\mathbb{N}\) the set of positive natural numbers, that is, \(\mathbb{N}=\{1,2,\dots \}\). For \(n\in \mathbb{N}\), let
Theorem 1.1
(Dragomir–Goşa [7])
Let \((X,d)\) be a metric space. Then, for all \(n\in \mathbb{N}\), \(n\geq 2\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\),
Moreover, the inequality is optimal in the sense that the multiplicative coefficient \(C=1\) on the right-hand side of (1.1) (in front of inf) cannot be replaced by a smaller real number.
In the particular case where \(p_{i}=\frac{1}{n}\) (\(i=1,2,\dots ,n\)), (1.1) reduces to
This inequality can be interpreted as follows. Let P be a polygon in a metric space with n vertices, and let x be an arbitrary point in the space. Then the sum of all edges and diagonals of P is less than n times the sum of the distances from x to the vertices of P.
In the same reference [7] the authors provided some interesting applications of inequality (1.1) to normed linear spaces and pre-Hilbert spaces. For more results on metric inequalities, we refer to [1, 6, 12] and the references therein.
In this paper, we derive new inequalities in 2-metric spaces and 2-normed linear spaces. In particular, we obtain an extension of Theorem 1.1 to the setting of 2-metric spaces and provide a geometric interpretation of the obtained inequality.
Before stating and proving our results, let us recall briefly some basic notions related to 2-metric spaces and 2-normed linear spaces.
In 1963, Gähler [10] introduced the notion of 2-metric spaces as follows. Let X be a nonempty set, and let \(D: X\times X\times X\to \mathbb{R}\). We say that D is a 2-metric on X if the following conditions are satisfied:
- (\(D_{1}\)):
-
for all \(x,y\in X\) with \(x\neq y\), there exists \(z=z(x,y)\in X\) such that
$$ D(x,y,z)\neq 0; $$ - (\(D_{2}\)):
-
\(D(x,y,z)=0\) when at least two elements of \(\{x,y,z\}\subset X\) are equal;
- (\(D_{3}\)):
-
for all \(x,y,z\in X\),
$$ D(x,y,z)=D(x,z,y)=D(y,z,x); $$ - (\(D_{4}\)):
-
for all \(x,y,z,u\in X\),
$$ D(x,y,z)\leq D(u,y,z)+D(x,u,z)+D(x,y,u). $$
In this case, the pair \((X,D)\) is called a 2-metric space.
Let us mention some remarks following from properties (\(D_{1}\))–(\(D_{4}\)).
-
Given \(x,y,z\in X\), we denote by \(\sigma (x,y,z)\) any permutation of the elements x, y, and z. By (\(D_{3}\)) we deduce that
$$ D(x,y,z)=D\bigl(\sigma (x,y,z)\bigr),\quad x,y,z\in X. $$ -
Let \(x,y,z\in X\). By (\(D_{3}\)) and (\(D_{4}\)), for all \(u\in X\), we have
$$ \begin{aligned} & D(x,y,z) \\ &\quad \leq D(u,y,z)+D(x,u,z)+D(x,y,u) \\ &\quad \leq D(x,y,z)+D(u,x,z)+D(u,y,x)+D(x,u,z)+D(x,y,u) \\ &\quad =D(x,y,z)+2D(u,x,z)+2D(u,y,x), \end{aligned} $$which yields
$$ D(u,x,z)+D(u,y,x)\geq 0. $$Taking \(u=y\) in this inequality and using (\(D_{2}\)), we obtain
$$ D(x,y,z)\geq 0,\quad x,y,z\in X. $$
Example 1.1
(see [10])
Let \(D: \mathbb{R}^{N}\times \mathbb{R}^{N}\times \mathbb{R}^{N}\to \mathbb{R}\), \(N\in \mathbb{N}\), \(N\geq 2\), be the mapping defined by
where × denotes the cross product in \(\mathbb{R}^{N}\), and \(\|\cdot \|_{2}\) denotes the Euclidean norm in \(\mathbb{R}^{N}\). Then D is a 2-metric on \(X=\mathbb{R}^{N}\). Note that \(D(A_{1},A_{2},A_{3})\) is equal to the area of the triangle spanned by \(A_{1}\), \(A_{2}\), and \(A_{3}\).
In the same reference [10], Gähler introduced the notion of 2-normed linear spaces as follows. Let X be a linear space over \(\mathbb{R}\) of dimension \(1< L\leq \infty \). Let \(\|\cdot ,\cdot \|: X\times X\to \mathbb{R}\) be a given mapping. We say that \(\|\cdot ,\cdot \|\) is a 2-norm on X if the following conditions are satisfied for all \(x,y,z\in X\) and \(\lambda \in \mathbb{R}\):
- (\(N_{1}\)):
-
\(\|x,y\|=0\) if and only if x and y are linearly dependent;
- (\(N_{2}\)):
-
\(\|x,y\|=\|y,x\|\);
- (\(N_{3}\)):
-
\(\|\lambda x,y\|=|\lambda |\|x,y\|\);
- (\(N_{4}\)):
-
\(\|x,y+z\|\leq \|x,y\|+\|x,z\|\).
In this case, the pair \((X,\|\cdot ,\cdot \|)\) is said to be a 2-normed space.
We now give some remarks following from (\(N_{1}\))–(\(N_{4}\)):
-
By (\(N_{2}\)) and (\(N_{3}\)), for all \(x,y\in X\) and \(\lambda ,\mu \in \mathbb{R}\), we have
$$ \Vert \lambda x,\mu y \Vert = \vert \lambda \vert \vert \mu \vert \Vert x,y \Vert = \Vert \mu x,\lambda y \Vert . $$ -
If \(\|\cdot ,\cdot \|\) is a 2-norm on X, then the mapping \(D: X\times X\times X\to \mathbb{R}\) defined by
$$ D(x,y,z)= \Vert x-z,y-z \Vert ,\quad x,y,z\in X, $$(1.3)is a 2-metric on X. Note that if \(L=1\), then condition (\(D_{1}\)) is not satisfied by D. Namely, by (\(N_{1}\)), if \(X=\operatorname{span}\{a\}\), \(a\in X\), then for all \(x,y,z\in X\), there exist \(\lambda ,\mu ,\gamma \in \mathbb{R}\) such that
$$ D(x,y,z)=D(\lambda a,\mu a,\gamma a)= \bigl\Vert (\lambda -\gamma )a,(\mu - \gamma )a \bigr\Vert = \bigl\vert (\lambda -\gamma ) (\mu -\gamma ) \bigr\vert \Vert a,a \Vert =0. $$ -
From the above remark and the positivity of D we deduce that
$$ \Vert x,y \Vert \geq 0,\quad x,y\in X. $$ -
Let \(x,y,z\in X\) and \(\lambda _{1},\lambda _{2}\in \mathbb{R}\). By (\(N_{2}\)) and (\(N_{4}\)) we have
$$\begin{aligned} \Vert \lambda _{1}x+\lambda _{2}y,z \Vert =& \Vert z, \lambda _{1}x+\lambda _{2}y \Vert \\ \leq & \Vert z,\lambda _{1} x \Vert + \Vert z,\lambda _{2} y \Vert \\ =& \vert \lambda _{1} \vert \Vert x,z \Vert + \vert \lambda _{2} \vert \Vert y,z \Vert . \end{aligned}$$Hence by induction we deduce that if \(x_{i},z\in X\) and \(\lambda _{i}\in \mathbb{R}\), \(i=1,2,\dots ,m\), then
$$ \Vert \lambda _{1}x_{1}+\lambda _{2}x_{2}+\cdots +\lambda _{m}x_{m},z \Vert \leq \sum_{i=1}^{m} \vert \lambda _{i} \vert \Vert x_{i},z \Vert . $$(1.4)
For more details about 2-metric spaces and 2-normed linear spaces, see, for example, [2–5, 8, 9, 11, 13–17] and the references therein.
2 Results and proofs
In this section, we state and prove our main results and provide some interesting consequences.
Theorem 2.1
Let \((X,D)\) be a 2-metric space. Then, for all \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\),
Moreover, the inequality is optimal in the sense that the multiplicative coefficient \(C=1\) on the right-hand side of (2.1) (in front of inf) cannot be replaced by a smaller real number.
Proof
Let \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\). Let x be an arbitrary element of X. For all \(i,j,k\in \{1,2,\dots ,n\}\), we have
Multiplying this inequality by \(p_{i}p_{j}p_{k}\) and taking the sum from 1 to n, we obtain
where
and
Sine \(\sum_{i=1}^{n} p_{i}=1\), by the symmetry of D we deduce that
On the other hand, by (\(D_{2}\))–(\(D_{3}\)) we have
that is,
Similarly, we have
that is,
Hence, using (2.2), (2.3), (2.4), and (2.5), we obtain
Since this inequality holds for all \(x\in X\), we deduce (2.1).
Suppose now that there exists a constant \(C>0\) such that
for all \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\). Taking \(n=3\) in (2.6), we obtain
for all \((p_{1},p_{2},p_{3})\in \Pi _{3}\), \(\{x_{i}\}_{i=1}^{3}\subset X\), and \(x\in X\). In particular, for \(x=x_{1}\) and \((p_{1},p_{2},p_{3})=(2\varepsilon -1,1-\varepsilon ,1-\varepsilon )\), \(\frac{1}{2}<\varepsilon <1\), by (\(D_{2}\)) we obtain
which yields
Passing to the limit as \(\varepsilon \to 1^{-}\), we get that \(C\geq 1\), which proves the sharpness of (2.1). □
Corollary 2.1
Let \((X,D)\) be a 2-metric space. Then, for all \(n\in \mathbb{N}\), \(n\geq 3\), and \(\{x_{i}\}_{i=1}^{n}\subset X\),
Proof
By (2.1) with
(2.7) follows. □
Corollary 2.1 has the following geometric interpretation.
Corollary 2.2
Let \(n\in \mathbb{N}\), \(n\geq 3\), and let \(A_{1},A_{2},\dots ,A_{n}, A\) be \(n+1\) points of \(\mathbb{R}^{N}\), \(N\geq 2\). Then the sum of the areas of all triangles with vertices belonging to the set of points \(\{A_{i}:\, i=1,2,\dots ,n\}\) is less than n times the sum of the areas of all triangles such that one of the vertices is the point A and the other vertices belong to the set of points \(\{A_{i}:\, i=1,2,\dots ,n\}\).
Proof
The result follows immediately from Corollary 2.1 by taking \(X=\mathbb{R}^{N}\) and D, the 2-metric defined by (1.2). □
Corollary 2.3
Let \((X,D)\) be a 2-metric space, \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\). Let \(x\in X\) be such that
for some \(r>0\). Then
Proof
By (2.1) we have
On the other hand, using (2.8), we obtain
Combining (2.10) with (2.11), (2.9) follows. □
Corollary 2.4
Let X be a linear space over \(\mathbb{R}\) of dimension \(1< L\leq \infty \), and let \(\|\cdot ,\cdot \|\) be a 2-norm on X. Then, for all \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\),
Moreover, the inequality is optimal in the sense that the multiplicative coefficient \(C=1\) on the right-hand side of (2.12) (in front of inf) cannot be replaced by a smaller real number.
Proof
Consider the 2-metric D on X defined by (1.3). Then (2.12) follows by (2.1). □
Theorem 2.2
Let X be a linear space over \(\mathbb{R}\) of dimension \(1< L\leq \infty \), and let \(\|\cdot ,\cdot \|\) be a 2-norm on X. Then, for all \(n\in \mathbb{N}\), \(n\geq 3\), \((p_{1},p_{2},\dots ,p_{n})\in \Pi _{n}\), and \(\{x_{i}\}_{i=1}^{n}\subset X\),
where
Proof
Using (2.12) with \(x=x_{p}\), we obtain
By (2.5) we have
On the other hand, using (\(N_{2}\)), we obtain
Next, by (1.4) we have that
Hence it follows from (2.15), (2.16), and (2.17) that
Finally, (2.13) follows from (2.14) and (2.18). □
For our next result, we need some notations.
Given three points \(A,B,C\in \mathbb{R}^{N}\), \(N\geq 2\), we denote by \(\bigtriangleup (A,B,C)\) the area of the triangle with vertices A, B, and C.
Let \(n\in \mathbb{N}\), \(n\geq 3\). For n points \(A_{1},A_{2},\dots , A_{n}\in \mathbb{R}^{N}\), let
We introduce the set
and the quantity
Theorem 2.3
For all \(n\in \mathbb{N}\), \(n\geq 3\), we have that \(\alpha _{n} \geq \frac{n}{18}\).
Proof
First, for all \(A,B,C\in \mathbb{R}^{N}\), we have
where D is the 2-metric defined by (1.2). On the other hand, given \(\{A_{1},A_{2},\dots ,A_{n}\}\in \Lambda _{n}\), for all \(j\in \{1,2,\dots ,n\}\), by (\(D_{4}\)), we have
for all \(P\in \{A_{1},A_{2},\dots ,A_{n}\}\). Taking the sum over j from 1 to n, we get that
that is,
Notice that
Hence by (2.19) we obtain
On the other hand, we have
and
Therefore, using (2.20), (2.21), and (2.22), we get that
Next, taking the sum over \(P\in \{A_{1},A_{2},\dots ,A_{n}\}\), we obtain
Notice that by (2.5) we have
Combining (2.23) with (2.24), we deduce that
which yields the desired estimate. □
3 Conclusion
We obtained new inequalities in the setting of 2-metric spaces and 2-normed linear spaces. Namely, we first derived an analogous version of Theorem 1.1 for 2-metric spaces (see Theorem 2.1). Moreover, we provided a geometric interpretation of our obtained result (see Corollary 2.2). We also presented some interesting consequences following from Theorem 2.1. Next, we considered a problem related to the estimates of areas of triangles and derived a new inequality (see Theorem 2.3).
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The second author is supported by Researchers Supporting Project number (RSP-2021/4), King Saud University, Riyadh, Saudi Arabia.
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Jleli, M., Samet, B. On some inequalities in 2-metric spaces. J Inequal Appl 2021, 130 (2021). https://doi.org/10.1186/s13660-021-02662-3
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DOI: https://doi.org/10.1186/s13660-021-02662-3