Simultaneously proximinal subspaces

In this paper we study closed subspaces of Banach spaces that admit relative Chebyshev centers for all bounded subsets of the space. We exhibit new classes of spaces which have this property and study stability results similar to the ones studied in the literature in the context of proximinal subspaces and Chebyshev centers. For the space of continuous functions on a compact set , we show that a closed subspace of finite codimension has relative Chebyshev centers for all bounded sets in if and only if it is a strongly proximinal subspace.