On weak<SUP>∗ </SUP>-extreme points in Banach spaces

We study the extreme points of the unit ball of a Banach space that remain extreme when considered, under canonical embedding, in the unit ball of the bidual. We give an example of a strictly convex space whose unit vectors are extreme points in the unit ball of the second dual but none are extreme points in the unit ball of the fourth dual. For the space of vector- valued continuous functions on a compact set we show that any function whose values are weak∗ -extreme points is a weak∗ -extreme point . We explore the relation between weak∗ -extreme points and the dual notion of very smooth points. We show that if a Banach space X has a very smooth point in every equivalent norm then X∗ has the Radon-Nikodaeym property