A note on extreme points in dual spaces (Q1940875)

Let \(X\) be a Banach space. In this short paper, the authors consider the question of when all extreme points of the unit ball of the triple dual of \(X\), when restricted to \(X\) (under the canonical embedding), are also extreme points of the dual unit ball of \(X\). They show that when \(X^\ast\) is isometric to an \(L^1(\mu)\)-space, this happens only when the extreme points of the dual unit ball are a weak\(^\ast\)-closed set.