On Singletonness of Remotal and Uniquely Remotal Sets

Abstract

A bounded subset $T$ of a metric space $(X,\rho)$ is said to be remotal (uniquely remotal) if for each $x\in X$ there exists at least one (exactly one) $t\in T$ such that $\rho(x,t)=\sup\{\rho(x,y):y\in T\}.$ Such a point $t$ is called a farthest point to $x$ in $T$. In this paper, we discuss properties of remotal and uniquely remotal sets and, conditions under which remotal and uniquely remotal sets are singleton. The underlying spaces are convex metric spaces or externally convex metric spaces.