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 find anything about it. I've asked this question on StackExchange as well.

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.