Let $X$ be a Banach space where the closed unit ball equals the convex hull of its extreme points. Is it true that this implies $X$ is reflexive?



A Banach space where the closed unit ball is the convex hull of its extreme points
The answer is "No" because there exist nonreflexive Banach spaces in which every point on the surface of the unit ball is extreme. See, for example, Diestel, Geometry of Banach spaces, Chapter 4, Section 2, Theorem 1 and apply it to the natural embedding of $\ell\_1$ into $\ell\_2$. 


