Normal structure and proximal normal structure

In 2005 Eldred et al. Introduced a notion called 'proximal normal structure' to study the existence of best proximity points for relatively nonex-pansive maps. In this paper we will give a characterization for proximal normal structure. Using this characterization we will prove that if A or B is compact, then the convex pair (A, B) has proximal normal structure. We will also show that in general if (A, B) is a closed bounded convex proximal pair, then the compactness of A need not imply the compactness of B. But in in k-strictly convex Banach spaces this is not the case. In addition we will use our characterization to prove the following: (i) If X is a Banach space and the set of all directions in which X is not uniformly convex is contained in a countable union of k-dimensional subspaces of X for some k ∈ or the set of all directions in which X is not uniformly convex is contained in a subspace of X with countable Hamel basis, then X has proximal normal structure. (II) Every nearly uniformly convex Banach space has proximal normal structure. (III) If (X, ∥·∥) is a Hilbert space and ∣·∣ a norm on X and 1 ≤ β < √2 such that (1/β)∥x∥ ≤∥x∥ ≤ ∣x∣ for all x ∈ X, then (X, ∣·∣) has proximal normal structure.