On Farthest Points

Abstract

A non-empty bounded subset T of  a metric space (X, d) is said to be remotal (uniquely remotal) if for each  x∈ X, there exists at least one (exactly one) t∈ T  such that d(x, t) = sup{d(x, y): y ∈T}, such a point t is called a farthest point of x in  T. In this paper, we discuss remotal sets, uniquely remotal sets and the singleton property of uniquely remotal sets, thereby providing some partial affirmative answers to the hitherto unsolved farthest point problem: If every point of a normed linear space X admits a unique farthest point in the set T, must T be a singleton? The underlying spaces considered are metric spaces, convex metric spaces, and linear metric spaces.