A Banach space where the closed unit ball is the convex hull of its extreme points

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?

• The closed unit ball of $\ell_1$ is the closed convex hull of its extreme points. – M.González Sep 26 '19 at 15:04
• @M.González: I think the point of the question is that the OP considers the convex hull of the extreme points without taking the closure. – Jochen Glueck Sep 26 '19 at 15:11
• lol at Banch space... – WhatsUp Sep 26 '19 at 15:23
• Oh yes, I didn't see the misprint in the title! Apologies. – Mark Roelands Sep 26 '19 at 15:34

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 $$\ell_2$$.
• @August Cleaner I don't understand your answer. By reflexive I mean that the canonical embedding $J$ from $X$ into its bidual $X^{**}$ is surjective, not that a norm deformation of the space yields something reflexive (this changes the entire geometry of the ball). Also, the closed unit ball of $\ell_1$ is not the convex hull of its extreme points. – Mark Roelands Sep 27 '19 at 8:28