On existence of farthest points (Q1070485)

Let K be a subset of a Banach space X with the property that every maximising sequence in K has a subsequence weakly converging to some element of K. Then the set \(\{x\in X:\| x-z\| =\sup_{y\in K}\| x-y\|\) for some \(z\in K\}\) contains a dense \(G_{\delta}\)-subset of X. This is an extension of a result of \textit{Ka Sing Lau} in Isr. J. Math. 22, 168-174 (1975; Zbl 0325.46022). Further, if X also satisfies Efimov Stechkin property, then the author shows that the above result is also valid for a set K which is simply closed and bounded.