Abstract. Brodskii and Milman proved that there exists a point in $C(A)$, the set of all Chebyshev centers of $A$, which is fixed by every surjective isometry from $A$ into $A$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. Motivated by this result, Lim et al. proved that every isometry from $A$ into $A$ has a fixed point in $C(A)$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. In this paper, we prove that every relatively isometry map $T\colon A\cup B \rightarrow A\cup B$, satisfying $T(A) \subseteq B$ and $T(B) \subseteq A$, has a best proximity point in $C_{A}(B)$, the set of all Chebyshev centers of $B$ relative to $A$, whenever the nonempty weakly compact convex proximal pair $(A, B)$ has proximal normal structure and rectangle property. Also, we prove that, under suitable assumptions, an analogous result of Brodskii and Milman for relatively isometry mappings holds. In case of $A = B$, we obtain the results of Brodskii and Milman, and Lim et al. as a particular case of our results.

DOI: 10.14232/actasm-014-833-1

AMS Subject Classification (1991): 47H09, 47H10

Keyword(s): asymptotic center, Chebyshev center, best proximity points, proximal pairs, relatively nonexpansive maps, rectangle property

Received December 30, 2014. (Registered under 83/2014.)