Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an example of a reflexive Banach space $X$ whose unit ball does not equal the convex hull of its extreme points? Such an example $X$ must be infinite-dimensional and can't be strictly convex. My feeling is that this should be known, but I couldn't find anything about it. I've asked this question on StackExchange as well.



Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?
I'd try $X:=\ell\_2$ with an equivalent but non strictly convex norm.


Let $(e\_k)\_{k\ge0}$ be the standard Hilbert basis of $\ell\_2$. Consider the sets:


* $A:=\{0\}\cup\{2^{-k}e\_k:k\ge 1\}$,
* $B$, the closed unit ball of $\ell\_2$,
* $C:=\overline{\text{co}}\, A$, a compact subset of the hyperplane $e\_0^\perp$.
* $D:=\overline{\text {co}}\big(B\cup( e\_0+C)\cup(-e\_0-C)\big)$.


Clearly, $C:=\big\{\sum\_{k\ge1}2^{-k}a\_ke\_k: a\_k \ge0 \text{ for all } k\ge1 \text{ and } \sum\_{k\ge1}a\_k\le1\big\},$ so $\text{ext}\, C=A$ and ${\text{co}}(\text{ext}\, C)={\text{co}}\, A\subsetneq C. $


The set $D$ is a bounded, closed, convex, symmetric nbd of $0$ in $\ell\_2$, so it is the closed unit ball in an equivalent norm of $\ell\_2$.


We have $D\cap (e\_0+e\_0^\perp)=e\_0+C;$ the set of extremal points of $D$ in the affine hyperplane $e\_0+e\_0 ^\perp$ is exactly the set $e\_0+A$, which implies that ${\text{co}}(\text{ext} D)\subsetneq D$ because the trace of these sets on $e\_0+ e\_0 ^\perp$ are ${\text{co}}(\text{ext} D)\cap (e\_0+ e\_0 ^\perp) =e\_0+\text{co}A\subsetneq D \cap (e\_0+ e\_0 ^\perp) $.


