Research Article Ordered Convex Metric Spaces Ismat Beg Department of Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan Correspondence should be addressed to Ismat Beg; ibeg@lahoreschool.edu.pk Received 30 August 2021; Revised 14 October 2021; Accepted 15 October 2021; Published 25 October 2021 Academic Editor: Alberto Fiorenza 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. 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 proper- ties. 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 nor- med/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 met- ric 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 gen- eral topology. In Section 3, we introduce the new concept of ordered convex metric spaces and study some basic proper- ties. Several characterizations of these spaces are also proven that allow making geometric interpretations of the new con- cepts 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 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, du, ω a, b ; λ ð Þ ð Þ ≤ λdu, a ð Þ +1 − λ ð Þdu, b ð Þ: ð1Þ 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 . Remark 2 (see [5, 10]). The convex metric space ðX, dÞ has the following properties: (i) wða, b ;1Þ = a, ωða, b ;0Þ = b, ωða, a ; λÞ = a (ii) Open spheres Bða, rÞ = fb ∈ X : dða, bÞ < rg and closed spheres B½a, r = fb ∈ X : dða, bÞ ≤ rg are convex (iii) If fK α : α ∈ Ag is a family of convex subsets of X, then ∩ K α α∈A is convex Any normed space and a convex subset of a normed space is a convex metric space. There are several examples Hindawi Journal of Function Spaces Volume 2021, Article ID 7552451, 4 pages https://doi.org/10.1155/2021/7552451