On some inequalities in 2-metric spaces

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.


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 N the set of positive natural numbers, that is, N = {1, 2, . . . }. For n ∈ N, let n = (p 1 , p 2 , . . . , p n ) ∈ R n : p i ≥ 0 (i = 1, 2, . . . , n), n i=1 p i = 1 . Theorem 1.1 ) Let (X, d) be a metric space. Then, for all n ∈ N, n ≥ 2, (p 1 , p 2 , . . . , p n ) ∈ n , and {x i } n i=1 ⊂ 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 = 1 n (i = 1, 2, . . . , 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 × X × X → R. We say that D is a 2-metric on X if the following conditions are satisfied: 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 ∈ X, we denote by σ (x, y, z) any permutation of the elements x, y, and z.
Taking u = y in this inequality and using (D 2 ), we obtain D(x, y, z) ≥ 0, x, y, z ∈ X. Example 1.1 (see [10]) Let D : R N × R N × R N → R, N ∈ N, N ≥ 2, be the mapping defined by where × denotes the cross product in R N , and · 2 denotes the Euclidean norm in R N . Then D is a 2-metric on X = 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 R of dimension 1 < L ≤ ∞. Let ·, · : X × X → R be a given mapping. We say that ·, · is a 2-norm on X if the following conditions are satisfied for all x, y, z ∈ X and λ ∈ R: (N 1 ) x, y = 0 if and only if x and y are linearly dependent; In this case, the pair (X, ·, · ) is said to be a 2-normed space.
• If ·, · is a 2-norm on X, then the mapping D : 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 = span{a}, a ∈ X, then for all x, y, z ∈ X, there exist λ, μ, γ ∈ R such that • From the above remark and the positivity of D we deduce that • Let x, y, z ∈ X and λ 1 , λ 2 ∈ R. By (N 2 ) and (N 4 ) we have Hence by induction we deduce that if x i , z ∈ X and λ i ∈ R, i = 1, 2, . . . , m, then 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.

Results and proofs
In this section, we state and prove our main results and provide some interesting consequences.
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.
Passing to the limit as ε → 1 -, we get that C ≥ 1, which proves the sharpness of (2.1).

Corollary 2.1 Let (X, D) be a 2-metric space. Then, for all n ∈ N, n ≥ 3, and {x
Proof By (2.1) with

Corollary 2.4
Let X be a linear space over R of dimension 1 < L ≤ ∞, and let ·, · be a 2-norm on X. Then, for all n ∈ N, n ≥ 3, (p 1 , p 2 , . . . , p n ) ∈ n , and {x i } n i=1 ⊂ 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.
For our next result, we need some notations. Given three points A, B, C ∈ R N , N ≥ 2, we denote by (A, B, C) the area of the triangle with vertices A, B, and C.

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).