On Simultaneous Characterization of the Set of Elements of Good Approximation in Metric Spaces

Abstract

If W is a subset of a metric space $(X,d)$ then for a given $\varepsilon>0$, an element $y_0\in W$ is called a good approximation or $\varepsilon-$approximation for $x\in X$ if $d(x,y_0)\leq d(x,W)+\varepsilon.$ We denote by $P_{W,\varepsilon}(x)$ the set of all such $y_0\in W$ i.e. $P_{W,\; \varepsilon}(x)=\{y\in W:d(x,y)\leq d(x,W)+\varepsilon\}$. In particular, for $\varepsilon=0$ we get the set of all best approximations to $x$ in W. Given a subset M of W, what are the necessary and sufficient conditions in order that every element $y_0\in M$ is an element of good approximation to $x$ by the elements of W? The paper mainly deals with this problem of simultaneous characterization of elements of good approximation in metric spaces. The proved results extend and generalize several known results on the subject.