Let X be a closed subspace of c₀. We show that the metric projection onto any proximinal subspace of finite codimension in X is Hausdorff metric continuous, which, in particular, implies that it is both lower and upper Hausdorff semicontinuous.
In any dual space X*, the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm which is itself contained in the set NA of norm attaining functionals. We show that NA and SSD coincide if and only if every proximinal hyperplane of X is strongly proximinal, and that if QP and NA coincide then every finite codimensional proximinal subspace of X is strongly proximinal. Natural examples and applications are provided.
The notion of ball proximinality and the strong ball proximinality were recently introduced in [2]. We prove that a closed * subalgebra A of C(Q) is strongly ball proximinal in C(Q) and the metric projection from C(Q), onto the closed unit ball of A, is Hausdorff metric continuous and hence has continuous selection.
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