Uniquely Remotal Sets in Banach Spaces

The study of the continuity of the farthest point mapping for uniquely remotal sets has been used extensively in the literature to prove the singletoness of such sets. In this article, we show that the farthest point mapping is not continuous even if the set is remotal, rather than being uniquely remotal. Consequently, we obtain some generalizations of results concerning the singletoness of remotal sets. In particular, it is proved that a compact set admitting a unique farthest point to its center is a singleton, generalizing the well known result of Klee.