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).