Farthest point problem and M-compact sets

Abstract: In this paper we give an elementary proof of the fact that every uniquely remotal set is singleton in a finite dimensional strictly convex normed linear space. We show that if A is a uniquely remotal M-compact subset with the derived set of A non-empty then the derived set of A is M-compact and uniquely remotal. We also show that if A is a uniquely remotal M-compact set and the derived set of A is compact then A is singleton.