[Recall the Bishop-Phelps Theorem][1]. > **Bishop-Phelps Theorem:** Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$ Then the set $$\{e^*\in E^*: e^* \text{ attains its supremum on } B \}$$ is norm-dense in the dual $E^*.$ Does the theorem hold for extreme points? More precisely, > *Question:* Let $B\subseteq E$ be a bounded, closed and convex subset of a real Banach space $E.$ Is it true that the set $$\{e^*\in\text{ext} \left( B_{E^*} \right): e^*\text{ attains its supremum on }B\}$$ norm-dense in $\text{ext}\left( B_{E^*} \right)?$ If the there is an affirmative answer to my question, may I have reference? ____ *Notation:* We say that $e^*$ is an extreme point of $B_{E^*}$ if it cannot be expressed as midpoint of two elements from $B_{E^*}.$ Denote $\text{ext}\left( B_{E^*}\right)$ to be the set of all extreme points of $B_{E^*}.$ [1]: https://en.wikipedia.org/wiki/Bishop%E2%80%93Phelps_theorem