On Preserved and Unpreserved Extreme Points

An extreme point of the closed unit ball of a Banach space is said to be preserved if it is extreme of the closed unit ball of the bidual space; otherwise it is called unpreserved. The beginning of the present work takes the form of a survey on this topic, presenting some elementary facta about those concepts—usually with new proofs—and discussing in particular Katznelson’s solution to a Phelps’ question on preserved extreme points, not available, to our knowledge, in the literature. In a second part, some new results are presented. Since some of them depend on the concept of polyhedrality, we first review several results on this topic. Then we present Godun renorming theorem for the class of nonreflexive Banach spaces, and Morris renorming result—with a new proof—on separable Banach spaces containing a copy of \(c_0\). We show that, under some extra conditions—polyhedrality—a similar renorming, this time adding smoothness, can be defined ensuring strict convexity with all points in the unit sphere unpreserved extreme. We finalize this work by presenting what—to our knowledge—is the first nonseparable result of this kind for the natural class of the weakly compactly generated Banach spaces.