Let $X$ be a Banach space. Do we have result such as: $X$ is reflexive if and only if the set of the extreme points of its unit ball $B$ is exactly the unit sphere $S$.
Context: I am dealing with a Banach space between $L_1$ and $L_2$ and wondering about its reflexivity. I am defining the space in this post but I am also interested in a general result on extreme points and reflexivity.
This is far from being true. For one, every finite-dimensional Banach space is reflexive, but if you consider $\mathbb K^2$ for example with the $\ell^1$ or $\ell^\infty$ norm, then not every point in the unit sphere is an extreme point of the unit ball. On the other hand, every separable Banach space has a strictly convex renorming, i.e., an equivalent norm such that the extreme points of the unit ball are exactly the elements of the unit sphere. This gives you plenty of examples of non-reflexive Banach spaces for which every element of the unit sphere is an extreme point of the unit ball.
