Best approximation in metric spaces

Abstract

Let (X, d) be a metric space, and G be a closed subset of X.For x E X, let p(x,G) = inf{d(x, y) : y E G}. If the infimum is attained for all x E X, then G is called proximinal in X.The problem of priximinality of subsets in normed spaces has been studied by many authors.We refer mainly to the encyclopedia of Singer [10], and other references cited there, where the problem is treated in detail. Singer suggested the problem of proximinality in the so-called convex metric spaces.In metric linear spaces, many results on the proximinality problem were obtained in [1,2] and other papers cited in [2].Some results on proximinality in metric (but necessarity linear) spaces appeared in [6 and 7]. In [7], Busemann considered externally convex metric spaces (X, d) [4, p. 51], which satisfy the following conditions: In this paper we prove results which relate proximinality of sets to the metric structure in the space. We prove among other things that Busemann's result is true for a larger class of metric spaces. More results on convexity and Chebyshevity of sets are presented.In §1, we define and characterize M-spaces. In §2, we study the convexity of balls in relation to prominality of convex sets in M-spaces. Some results on proximinality in strictly convex metric spaces are obtained.

Fixed point, approximate fixed point and Kantorovich-Rubinstein maximum principle in convex metric spaces

“…It has been proved [20] that P M (x) is convex for a closed convex set M in a convex metric space X having property (A). Definition 14 ([1]) Let K be a closed bounded and convex subset of a convex metric space X, a self map T of K is called weakly contractive if for each x, y ∈ K,…”

“…Aronszajn and Panitchpakdi [3] and Menger [25] defined the convexity structure on metric spaces through closed balls and studied their properties. Khalil [20] further studied existence of fixed points and best approximation in these convex metric spaces (see also [19]). This paper focuses on the convexity structure of a metric space, using the geometric properties of this space; we study fixed points, approximate fixed points, structure of the set of fixed points and its application in Approximation theory.…”

Ordered Convex Metric Spaces

“…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].…”

On δ-suns

“…The metric space (X, d) together with a convex structure is called a convex metric space [8]. A convex metric space (X, d) is called an M-space [5] if for every two points x, y in X with d(x, y) = λ, and for every r ∈ [0, λ], there exists a unique z r ∈ X such that…”

“…An M-space (X, d) is called a strong M-space [5] if for every two points x, y in Every normed linear space is a strong M-space as well as an externally convex M-space but not conversely. If (X, d) is a convex metric space then for each two distinct points x, y ∈ X and for every λ, 0 λ 1, there exists at least one point…”