Some examples concerning proximinality in Banach spaces (Q498469)

Let \(X\) be a Banach space and denote by \(n\) and \(w\), respectively, the normal and the weak topology of \(X\). Let \(C\) be a subset of \(X\). The following relations are known: \[ \begin{tikzcd} \begin{matrix} C\text{ \(n\)-strongly}\\ \text{Chebyshev}\end{matrix} \ar[r] \ar[dd] & \begin{matrix} C\text{ approximatively}\\ \text{\(n\)-compact}\end{matrix} \ar[r] \ar[dd] & \begin{matrix} C\text{ \(n\)-strongly} \\ \text{proximinal } \end{matrix}\ar[dd] \ar[dr] & \\ &&& C\text{ proximinal }\\ \begin{matrix} C\text{ \(w\)-strongly}\\ \text{Chebyshev}\end{matrix} \ar[r] & \begin{matrix} C\text{ approximatively}\\ \text{\(w\)-compact}\end{matrix} \ar[r] & \begin{matrix} C\text{ \(w\)-strongly}\\ \text{proximinal}\end{matrix}\ar[ur] & \end{tikzcd} \] Moreover, \(C\) proximinal does not imply \(C\) \(n\)-(or \(w\)-)strongly proximinal. The main object of this paper is the study of the converse of the other relations above mentioned. Among other things, the authors show four counterexamples which prove that none of reverse of the above relations is generally true.