Best aproximation in metric spaces

Abstract

The notion of strict convexity and uniform convexity in normed linear spaces was extended to metric spaces in [1] and certain existence and uniqueness theorems on best approximation were proved in these spaces in [1] and [2] .In this note we shall give a relationship between the two types of convexities in metric spaces and further discuss some results on best approximation in metric spaces . We shall also extend the notion of sun introduced in normed linear spaces by Efimov and Steckin [31 to metric spaces .2 . Strictly-Convex And Uniformly-Convex Metr ic Spaces .If x,y,z are any these points in a metric space (X,d) then z is said to be a poínt between x and y if d(x,z) + d(z,y) = d(x,y) .

Strong Proximinality in Metric Spaces

“…The results proved in this paper generalize and extend some results of [3], [6], [7], [9], [12] and [15].…”

“…We also prove that in an M-space (X, d), if a Chebyshev set W is strongly proximinal at x, then W is strongly proximinal at every point between x and P W (x). The results proved in this section are motivated by the corresponding results proved for best approximation in Banach spaces given in [7] and in metric spaces given in [9] and [12]. Using the fact that a proximinal convex subset of a strictly convex metric space is Chebyshev (see [9]), we prove the following theorem: Theorem 3.1.…”

“…The authors do not know whether (iii)⇒ (ii) in Theorem 3.3. We require the following lemma proved in [12] for our next theorem. If W is strongly proximinal at x ∈ X, then W is strongly proximinal at every point between x and P W (x).…”

Proximinality and co-proximinality in metric linear spaces

“…G(R G ) = {(x, R G (x)) : x ∈ X}. Concerning the graph of R G , we have the following theorem (a similar result for the metric projection P G was proved in [6]).…”

“…Let X = R and G = (−1, 1) then −2 ∈ R −1 G (0), but −1 / ∈ R −1 G (0). (8) It is known (see [6]) that if G is a Chebyshev subset of a convex metric space (X, d), then P G (z) = P G (x), where z ∈ X is any element between x and P G (x). Does such a result hold for best coapproximation?…”