Best Constrained Approximation in Banach Spaces

Abstract

In this article, for a real Banach space X and a closed subspace Y, 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 for x ∉ Y, Y is constrained in span{x, Y}. We show that if X, under the canonical embedding has the strong--ball property in its bidual, then the same is true of Y. 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 Grothendieck G-space is a G-space.