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?

2$\begingroup$ The closed unit ball of $\ell_1$ is the closed convex hull of its extreme points. $\endgroup$ – M.González Sep 26 '19 at 15:04

2$\begingroup$ @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. $\endgroup$ – Jochen Glueck Sep 26 '19 at 15:11

1$\begingroup$ lol at Banch space... $\endgroup$ – WhatsUp Sep 26 '19 at 15:23

$\begingroup$ Oh yes, I didn't see the misprint in the title! Apologies. $\endgroup$ – 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 $\ell_2$.

7$\begingroup$ Indeed, any separable Banach space admits an equivalent norm that is strictly convex. And thus every point on the surface of the unit ball is an extreme point. $\endgroup$ – Gerald Edgar Sep 26 '19 at 19:35

$\begingroup$ @Gerald Edgar Is not it a strange thing to do: to (almost) repeat the posted answer adding "Indeed" before it (even if the question is simple)? $\endgroup$ – August Cleaner Sep 26 '19 at 20:21

5$\begingroup$ @AugustCleaner For someone like me, not having immediate access to Diestel's book, "almost any separable Banach space [does X]" adds nonzero information to "there exist nonreflexive Banach spaces [that doX]." $\endgroup$ – Andreas Blass Sep 26 '19 at 22:09


$\begingroup$ @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. $\endgroup$ – Mark Roelands Sep 27 '19 at 8:28