Ordered Convex Metric Spaces

Beg, Ismat

doi:https://doi.org/10.1155/2021/7552451

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 7552451 | https://doi.org/10.1155/2021/7552451

Ordered Convex Metric Spaces

Ismat Beg¹

Academic Editor: Alberto Fiorenza

Received30 Aug 2021

Revised14 Oct 2021

Accepted15 Oct 2021

Published25 Oct 2021

Abstract

The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts.

1. Introduction

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 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.

2. Preliminaries

In this section, basic results about convex metric spaces and order structure are given.

Definition 1 (see [5]). Let be a metric space and . A mapping is said to be a convex structure on if for each and ,

Metric space together with the convex structure is called a convex metric space. A nonempty subset is said to be convex if whenever .

Remark 2 (see [5, 10]). The convex metric space has the following properties: (i), , (ii)Open spheres and closed spheres are convex(iii)If is a family of convex subsets of , then is convex

Any normed space and a convex subset of a normed space is a convex metric space. There are several examples in the existing literature [5, 7, 8, 10] of convex metric spaces which are not embedded in any normed space.

Definition 3 (see [11]). A binary relation defined for some pairs of elements of a set is called an order relation in 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 . By we mean and Relation is defined as if and The inverse of is defined as if . Incomparable elements and (i.e., and ) are denoted by Transitivity of order relation implies for all .

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 be a connected topological space. If is a connected subset of such that is the union of nonempty, pairwise disjoint open (in ) sets , then is connected for all .

3. Ordered Convex Metric Spaces

In this section, first, we introduce the property 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.

Definition 8. A convex metric space is said to have property if for all in and in , we have

Each normed space has property , if we define In Definition 8, taking and using Remark 2, we obtain

Let be a convex metric space and be an ordered relation on . First, we define some notation for subsequent use. For any in and

Definition 9. (i) A relation on a convex metric space is said to be continuous if for all in , the sets and are closed.
(ii) A relation on a convex metric space is said to be Archimedean if for all in with , there exists such that and When relation is total, the space is called Archimedean.
(iii) A relation on a convex metric space is said to have betweenness property if for all in , all and all if and only if

Definition 10. A relation on a convex metric space , is (i)(o)-convex if and implies (ii)(o)-concave if and implies (iii)(o)-linear if implies

Definition 11. A convex metric space with order relation is called ordered convex metric space if is continuous.

Proposition 12. Let be an ordered convex metric space with property Then, is Archimedean and are open for any in

Proof. Let be an Archimedean relation on and be closed. Without loss of generality, we can assume that is nonempty. Choose in Now, continuity of and imply that there exists such that is contained in the complement of .
Assume there exists Continuity of implies that is an open set in Thus, 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 such that If , then set ; otherwise, Then, By Definition 8 (ii), we have Obviously, ; thus, It contradicts that is Archimedean. Hence, is open.
Similarly, we can prove that is open.
Assume that and are open. Choose in such that Remark 2 (i) implies that Since is open, thus there exists such that Also, and is open; therefore, there exists such that Now obviously, and
Next, we give Example 13 to show that we cannot drop any condition from Proposition 12.

Example 13. Consider the convex metric space with usual Euclidean distance and convex structure defined by Assume is a reflexive relation on such that for all and no other element are comparable. Then, each of and either contains at most two elements or is equal Therefore, and are closed. Also is not open. Moreover, but for all and all Thus, is continuous, and are not open, and is not Archimedean.

Now, the following proposition is obvious.

Proposition 14. Any totally ordered convex metric space with property is Archimedean.

Theorem 15. A nontrivial continuous Archimedean order on a convex metric space with property is totally ordered.

Proof. Let be not totally ordered on the convex metric space . Then, there exists such that and with Let Then, using Remark 4 or Since , therefore Thus, and Now, we prove Obviously, To prove other inclusions, choose If then it follows from transitivity and that , which is a contradiction to Therefore, , i.e., In a similar way, if then which contradicts Thus,
Now, and Remark 2 (i) imply that and Continuity of further implies that is closed. Using Equality 3, we obtain that is a closed set. On the other hand, Proposition 12 implies that is an open set. Thus, we have a nonempty closed-open proper subset of Since is connected, therefore it is a contradiction to Proposition 6 Similarly, we can show a contradiction for the case Hence, is a totally ordered relation.

Corollary 16. Let be an ordered convex metric space with property If is an Archimedean relation, then the space is also Archimedean.

Proposition 17. Let be an ordered convex metric space with property ; then, is (o)-linear is convex.

Proof. Let be (o)-linear. Choose and Define and Then, and Transitivity of implies It follows from (o)-linearity of that By Definition 8 (ii), we obtain Transitivity of further implies that Therefore, Hence, is convex.
Assume is convex. Choose such that and . From reflexivity of Remark 2 implies Since by assumption is convex. Therefore, Thus, Now transitivity of and imply that and Property (see Definition 8 (i)) of implies . Hence, is (o)-linear.

Proposition 18. Let be an ordered convex metric space with property , then is (o)-convex if and only if is convex.

Proof. Suppose that is (o)-convex. Choose and Define and Then, and (o)-convexity of implies that Using Definition 8, we have Thus, Hence, is convex.
Assume that is convex. Choose such that and Remark 2 implies As is convex, therefore Thus, Hence, is (o)-convex.

Proposition 19. Let be an ordered convex metric space with property ; then, is (o)-concave if and only if is convex.

Proof. Similar to Proposition 18.

Theorem 20. Let be an Archimedean-ordered convex metric space with property The relation is (o)-linear if and only if is (o)-convex and (o)-concave.

Proof. Assume is (o)-linear. Proposition 17 implies is convex, thus a connected subset of Obviously, Clearly, , and are mutually disjoint sets. Proposition 12 and continuity of imply that these all three sets are open sets. Since Proposition 18 and Proposition 19 further imply that (o)-concavity of is equivalent to the convexity of and (o)-convexity of is equivalent to the convexity of Now, Proposition 7 implies that is (o)-convex and (o)-concave.
Assume that is convex and is concave for all in . As Therefore, is convex. Proposition 17 further implies that is (o)-linear.

4. Concluding Remarks

Order, convexity, and metric are three fundamental concepts in mathematics. These ideas have beautiful geometric 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.

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Copyright

Copyright © 2021 Ismat Beg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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