On singletonness of remotal and uniquely remotal sets (Q2430157)

Let \(X\) be a normed space and \(E\) be a bounded set of \(X\). For \(x \in X\), let \(D(x; E) = \sup\{\| x - e \| : e \in E\}\). The set \(E\) is called remotal if there is some \(y \in E\) such that \(D(x, E) = \| x- y\| \). If \(y\) is unique for every \(x \in X\) then \(E\) is called uniquely remotal. One of the main conjectures in the theory of remotal sets is ``every uniquely remotal set in a normed space is a singleton''. In this paper, the authors study such conjecture and the concept of uniquely remotal sets in certain metric spaces.