For A bounded subset G of a metric Space (X,d) and $chi in X$, let $f_{G}$ be the real-valued function on X defined by $f_{G}$($chi$)=sup{$d (chi, g)in:G$}, and $F(G,chi)$={$z in X:sup_{g in G}d(g,z)=sup_{g in G}d(g,chi)+d(chi,z)$}. In this paper we discuss some properties of the map $f_G$ and of the set $ F(G, chi)$ in convex metric spaces. A sufficient condition for an element of a convex metric space X to lie in $ F(G, chi)$ is also given in this pope.