On the \(C\)-farthest points (Q2766068)

Let \(C\) be a norm-closed, convex set and \(0\in\text{int} C\) in a Banach space \(X\). Without loss of generality, one may assume that the unit ball of \(X\) is contained in \(C\). Denote by \(\rho\) the Minkowski functional of \(C:\rho(x): = \inf \{\lambda> 0:x\lambda^{-1}\in C\}\). For a bounded subset \(S\) in \(X\) and \(x \in X\) define the \(C\)-farthest distance \(x\) to \(S\) by \(\tau(S,x)= \sup\{\rho (x-y):y\in S\}\). In this situation, the author of the present note obtains the following theorem.NEWLINENEWLINENEWLINETheorem: Let \(X\) be a Banach space. Let \(S\) be a weakly compact subset in \(X\). Then the set \(\{x\in X:\rho(x-z) =\tau(S,x)\) for some \(z\in S\}\) contains a dense \(G_\delta\) of \(X\). Furthermore, the set of \(C\)-farthest points of \(S\) is nonempty.