Best Constrained Approximation in Banach Spaces.

In this article, for a real Banach spaceXand a closed subspaceY, we consider aspects of proximinality, its stronger variants and the notion of centrability, Chebyshev centers, for a class of subspaces that are relatively constrained in a Banach space, in the sense that forx ∉ Y,Yis constrained inspan{x,Y}. We show that ifX, under the canonical embedding has the strong--ball property in its bidual, then the same is true ofY. We also give applications of these results to proximinality in spaces of Bochner integrable functions. We consider a class of Banach spaces for which a formula due to Smith and Ward on relative Chebyshev centers and radius is valid. We show that any locally constrained subspace of the GrothendieckG-space is aG-space. [ABSTRACT FROM AUTHOR]