Proximal Normal Structure on Banach Spaces

Eldred, Kirk and Veeramani introduced the notion of proximal normal structure to prove the best proximity point theorems for relatively nonexpansive mappings and showed that uniformly convex Banach spaces have proximal normal structure. In this paper, a characterization for the weak proximal normal structure is given. Using this characterization, it is proved that every weakly compact convex pair in a Banach space X has proximal normal structure whenever X satisfies: X is ε0-inquadrate in every direction for some ε0∈(0,1) or X has the modulus of k-UC δXk(1)>0, for k∈ℕ or X has the modulus of k-dimensional U-convexity UXk(1)>0, for k∈ℕ or X has the coefficient of noncompact convexity ε1(X)<1. Moreover, we generalize the notion ε0-inquadrate in every direction to ε0-inquadrate with respect to every k-dimensional subspace and showed that X has the weak proximal normal structure and the weak normal structure if ε0∈(0,1). In the case of ε0∈[1,2), the Banach space X has weak proximal normal structure with an additional assumption X has the WORTH property.