Renorming Banach spaces with many projections and smoothness properties (Q1107789)

We consider the class of Banach spaces with a projectional separable resolution that have the same density character as their dual. We prove that such spaces have an equivalent norm such that the unit ball of the dual is the closed convex circled hull of a family of strongly exposed points with a uniformity in the strong exposition. Applications with smoothness properties are also given. We improve results of van Dulst- Singer and Finet-Schachermayer.