Some remarks on proximinality in higher dual spaces

In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}{α∈Γ} we show that any finite co-dimensional proximinal subspace of X=⊕c0Xα is strongly proximinal in all the duals of even order of X.