STRONG PROXIMINALITY AND ROTUNDITIES IN BANACH SPACES

Some necessary and sufficient conditions under which every proximinal convex subset of a non-reflexive Banach space X is strongly proximinal and JX(x∗) is compact for every norm attaining functional x∗ of SX∗ have been discussed. As a consequence, it is observed that if the conjugate space X∗ is strongly subdifferentiable for every norm attaining functional x∗ of SX∗ then X is nearly strongly rotund if and only if the metric projection onto every proximinal convex subset of X is upper semicontinuous. Some characterizations of non-reflexive strongly rotund Banach spaces have been discussed. Relationships between different types of rotundities and Property-(H), and examples to support these results have been given. It is also proved that a compactly locally uniform rotund Banach space is nearly strongly rotund and the converse holds if the space has Property-(WM).