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?
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2$\begingroup$ The closed unit ball of $\ell_1$ is the closed convex hull of its extreme points. $\endgroup$– M.GonzálezCommented Sep 26, 2019 at 15:04
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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 GlueckCommented Sep 26, 2019 at 15:11
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1$\begingroup$ lol at Banch space... $\endgroup$– WhatsUpCommented Sep 26, 2019 at 15:23
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$\begingroup$ Oh yes, I didn't see the misprint in the title! Apologies. $\endgroup$– Mark RoelandsCommented Sep 26, 2019 at 15:34
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1 Answer
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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 $\ell_2$.
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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. $\endgroup$ Commented Sep 26, 2019 at 19:35
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$\begingroup$ @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)? $\endgroup$ Commented Sep 26, 2019 at 20:21
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5$\begingroup$ @AugustCleaner For someone like me, not having immediate access to Diestel's book, "almost any separable Banach space [does X]" adds non-zero information to "there exist non-reflexive Banach spaces [that doX]." $\endgroup$ Commented Sep 26, 2019 at 22:09
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$\begingroup$ @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. $\endgroup$ Commented Sep 27, 2019 at 8:28