Existence of generalized nearest points (Q1811450)

Let \(X\) be a real Banach space of dimension at least \(2,\) and \(C\) a closed bounded convex subset of \(X\) with \(0\in \text{int } C,\) which is not necessarily symmetric. Let also \(P_{C}\) denote the Minkowski functional attached to the set \(C.\) If \(G\) is a nonempty closed subset of \(X\) and \(x\in X,\) then an element \(g_{0}\in G\) is called a generalized nearest point of \(x\) in \(G\) if \[ P_{C}( x-z_{0}) =d_{G}( x) :=\inf\{ P_{C}( x-z) :z\in G\} . \] The author studies the relations between the existence of generalized nearest points and the directional derivative of the function \(d_{G}.\) One shows that if \(d_{G}\) has the derivatives at \(x\) in any direction \(y\in\partial C\) (or \(-y\in\partial C\)) equal to \(1\) (resp. \(-1\)) then \(x\) has generalized nearest points in \(G.\) In the final part of the paper, one proves that if \(X\) is reflexive and \(C\) is strictly convex and has the Kadec-Klee property (\(\{ x_{n}\} \subset\partial C\) and \(x_{n}\to x_{0} \in\partial C\) weakly implies \(x_{n}\to x_{0}\) strongly), then the set \(\{ x\in X\setminus G:\exists y\in\partial C\) with \(d_{G}^{\prime }( x) ( y) =1\} \) is residual in \(X\setminus G.\)