On the size of the set of points where the metric projection exists (Q1885541)

A closed convex set \(C\) in a reflexive, locally uniformly convex Banach space \(X\) admits best approximations to a dense \(G_{\delta}\) subset of \(X\) [\textit{E. Asplund}, Isr. J. Math. 4, 213--216 (1966; Zbl 0143.34904)]. This work answers negatively a conjecture that if the hypothesis of the local uniform convexity is reduced to strict convexity then almost every \(x \in X\) has a best approximation in \(C\). A similar negative result is presented for farthest points and a conjecture that a property of differentials in reflexive spaces remains true for a non-reflexive space is shown to be false.