Two results on strong proximinality (Q2043211)

\textit{G.~Godefroy} and \textit{V.~Indumathi} [Rev. Mat. Complut. 14, No.~1, 105--125 (2001; Zbl 0993.46004)] proved that the kernel of a functional \(x^*\) is strongly proximinal if and only if the dual norm is strongly subdifferentiable at~\(x^*\); see [loc. cit.] or the present paper for unexplained notation. The author uses this characterisation to provide a criterion for a subspace \(Y\) of a Banach space \(X\) to be strongly proximinal and he uses it to give a quick proof (quick modulo [loc. cit.]) of the known result that \(M\)-ideals are strongly proximinal.