Ordered Convex Metric Spaces

Menger [1] initiated the study of convexity in metric spaces which was further developed by many authors [2–4]. The terms “metrically convex” and “convex metric space” are due to [2]. Subsequently, Takahashi [5] introduced the notion of convex metric spaces and studied their geometric properties. Takahashi also proved that all normed spaces and their convex subsets are convex metric spaces and gave an example of a convex metric space which is not embedded in any normed/Banach space. Kirk [6] showed that a metric space of hyperbolic type is a convex metric space. Afterward, Shimizu and Takahashi [7] gave the concept of uniformly convex metric space, studied its properties, and constructed examples of a uniformly convex metric space. Beg [8] established some inequalities in uniformly convex complete metric spaces analogous to the parallelogram law inHilbert spaces and their applications. Beg [9] proved that a closed convex subset of uniformly convex complete metric spaces is a Chebyshev set. Recently, Abdelhakim [10] studied convex functions on these spaces. The aim of this note is to further continue the research in this direction by introducing the concept of ordered convex metric spaces and study their structure. We conclude with the plan of the paper. In Section 2, we recall some basic notations and definitions from the existing literature on convex metric spaces, order structure, and general topology. In Section 3, we introduce the new concept of ordered convex metric spaces and study some basic properties. Several characterizations of these spaces are also proven that allow making geometric interpretations of the new concepts Finally, Section 4 concludes with a summary statement. 2. Preliminaries


Introduction
Menger [1] initiated the study of convexity in metric spaces which was further developed by many authors [2][3][4]. The terms "metrically convex" and "convex metric space" are due to [2]. Subsequently, Takahashi [5] introduced the notion of convex metric spaces and studied their geometric properties. Takahashi also proved that all normed spaces and their convex subsets are convex metric spaces and gave an example of a convex metric space which is not embedded in any normed/Banach space. Kirk [6] showed that a metric space of hyperbolic type is a convex metric space. Afterward, Shimizu and Takahashi [7] gave the concept of uniformly convex metric space, studied its properties, and constructed examples of a uniformly convex metric space. Beg [8] established some inequalities in uniformly convex complete metric spaces analogous to the parallelogram law in Hilbert spaces and their applications. Beg [9] proved that a closed convex subset of uniformly convex complete metric spaces is a Chebyshev set. Recently, Abdelhakim [10] studied convex functions on these spaces. The aim of this note is to further continue the research in this direction by introducing the concept of ordered convex metric spaces and study their structure.
We conclude with the plan of the paper. In Section 2, we recall some basic notations and definitions from the existing literature on convex metric spaces, order structure, and general topology. In Section 3, we introduce the new concept of ordered convex metric spaces and study some basic properties. Several characterizations of these spaces are also proven that allow making geometric interpretations of the new concepts Finally, Section 4 concludes with a summary statement.

Preliminaries
In this section, basic results about convex metric spaces and order structure are given.
Definition 1 (see [5]). Let ðX, dÞ be a metric space and I = ½0, 1. A mapping ω : X × X × I ⟶ X is said to be a convex structure on X if for each ða, b, λÞ ∈ X × X × I and u ∈ X, Metric space ðX, dÞ together with the convex structure ω is called a convex metric space. A nonempty subset K ⊂ X is said to be convex if ωða, b ; λÞ ∈ K whenever ða, b, λÞ ∈ K × K × I.
Definition 3 (see [11]). A binary relation a≼b defined for some pairs a, b of elements of a set X is called an order relation in X if ≼ is reflexive, transitive, and antisymmetric. A reflexive and transitive relation ≼ is called a preorder.
Remark 4 (see [11]). Let ≼ be a binary relation on a set X. By a ≺ b we mean a≼b and a ≠ b: Relation~is defined as a~b if a≼b and b≼a: The inverse of ≼ is defined as a ≽ b if b≼a. Incomparable elements a and b (i.e., a ≰ b and a ≱ b) are denoted by a⊳⊲b: Definition 5 (see [11]). An ordered set is called totally ordered if it has no incomparable elements.

Proposition 6 (see Proposition 4.1 of [12]). A topological space is disconnected if and only if it has a nonempty subset that is both open and closed.
Proposition 7 (see [13]). Let X be a connected topological

Ordered Convex Metric Spaces
In this section, first, we introduce the property ðLÞ on a convex metric space. Next, we present some notations and definitions related to an order relation ≼ on a convex metric space. Finally, we define ordered convex metric space and prove several interesting results related to ordered convex metric spaces.
Each normed space has property ðLÞ, if we define ω ða, b ; tÞ = ta + ð1 − tÞb: In Definition 8, taking μ = 0 and using Remark 2, we obtain Let X be a convex metric space and ≼ be an ordered relation on X. First, we define some notation for subsequent use. For any a, b, c in X and λ ∈ I, Definition 9. (i) A relation ≼ on a convex metric space X is said to be continuous if for all a, b, c in X, the sets A ≼ ða, b, cÞ and A ≽ ða, b, cÞ are closed.
(ii) A relation ≼ on a convex metric space X is said to be Archimedean if for all a, b, c, d in X with d⊳⊲a ≺ b⊳⊲c, there exists λ, μ ∈ ð0, 1Þ such that ωða, c ; λÞ ≺ b and a ≺ ωðb, d ; μÞ: When relation ≼ is total, the space is called Archimedean.
(iii) A relation ≼ on a convex metric space X is said to have betweenness property if for all a, b in X, all c ∈ fa, bg and all λ ∈ ð0, 1Þ,a≼b if and only if ωða, c ; λÞ≼ωðb, c ; λÞ: Proof. Let ≼ be an Archimedean relation on X and A ≺ ða, b, cÞ be closed. Without loss of generality, we can assume that A ≺ ða, b, cÞ is nonempty. Choose λ in A ≺ ða, b, cÞ: Now, continuity of ≼ and λ ∉ A ≽ ða, b, cÞ imply that there exists t > 0 such that N t ðλÞ = fα : |α − λ|<tg is contained in the complement of A ≽ ða, b, cÞ.
Assume there exists β ∈ N t ðλÞ ∩ A ⊳⊲ ða, b, cÞ: Continuity of ≼ implies that A ⊳⊲ ða, b, cÞ is an open set in I: Thus, A ⊳⊲ ða, b, cÞ is union of at most countably many mutually disjoint open intervals. Axiom of choice further implies that there exists among these intervals an open interval F such that β ∈ F: If β ≺ λ, then set δ = sup F; otherwise, δ = inf F: Then, δ ∈ A ≺ ða, b, cÞ: By Definition 8 (ii), we have Proof. Let ≼ be not totally ordered on the convex metric space X. Then, there exists u, v ∈ X such that u⊳⊲v and a, b ∈ X with a ≺ b: Let a ≺ u: Then, using Remark 4 a ≺ v or v ≺ u: Since u⊳⊲v, therefore a ≺ v: Thus, a ≺ u and a ≺ v: Now, we prove Obviously, A ≺ ða, u, vÞ ∩ A ≺ ða, u, uÞ ⊂ A ≼ ða, u, vÞ ∩ A ≼ ða, u, uÞ: To prove other inclusions, choose λ ∈ A ≼ ða, u, vÞ ∩ A ≼ ða, u, uÞ: If v~ωða, u ; λÞ, then it follows from transitivity and ωða, u ; λÞ≼u that u ≽ v, which is a contradiction to u⊳ ⊲v: Therefore, v ≻ ωða, u ; λÞ, i.e., λ ∈ A ≺ ða, u, vÞ: In a similar way, if u~ωða, u ; λÞ, then u≼v which contradicts u⊳⊲v: Thus, λ ∈ A ≺ ða, u, uÞ: Now, u⊳⊲v and Remark 2 (i) imply that 0 ∉ A ≺ ða, u, vÞ ∩ A ≺ ða, u, uÞ and 1 ∈ A ≺ ða, u, vÞ ∩ A ≺ ða, u, uÞ: Continuity of ≼ further implies that A ≼ ða, u, uÞ ∩ A ≼ ða, u, vÞ is closed. Using Equality 3, we obtain that A ≺ ða, u, vÞ ∩ A ≺ ða, u, uÞ is a closed set. On the other hand, Proposition 12 implies that A ≺ ða, u, uÞ ∩ A ≺ ða, u, vÞ is an open set. Thus, we have a nonempty closed-open proper subset of I: Since I is connected, therefore it is a contradiction to Proposition 6 : Similarly, we can show a contradiction for the case a ≻ b: Hence, ≼ is a totally ordered relation.
Corollary 16. Let X be an ordered convex metric space with property ðLÞ: If ≼ is an Archimedean relation, then the space X is also Archimedean.

Concluding Remarks
Order, convexity, and metric are three fundamental concepts in mathematics. These ideas have beautiful geometric 3 Journal of Function Spaces properties with significant applications in approximation and optimization (see [14,15]). In this work, we tried to combine these three indispensable notions of order, convexity, and metric. We introduced the new concept of ordered convex metric spaces and studied some of their properties. Several characterizations (Propositions 12, 17, 18, and 19 and Theorem 20) of these spaces are proven that allow to make geometric interpretations of the new concepts. This author's recommendation is to study other applications of ordered convex metric spaces to economics, preference modelling, control theory, functional analysis, etc.

Data Availability
No data were used to support this study.

Conflicts of Interest
Author declares that he has no conflict of interest.