  Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A
\cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq
d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a
mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In
this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P.
Veeramani, Proximal normal structure and relatively nonexpansive mappings,
Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in
Banach spaces to more general metric spaces. Namely, we give existence results
of best proximity points for cyclic and noncyclic relatively nonexpansive
mappings in the context of Busemann convex reflexive metric spaces. Moreover,
particular results are proved in the setting of CAT(0) and uniformly convex
geodesic spaces. Finally, we show that proximal normal structure is a
sufficient but not necessary condition for the existence in $A \times B$ of a
pair of best proximity points.
