[Recall the Bishop-Phelps Theorem](https://en.wikipedia.org/wiki/Bishop%E2%80%93Phelps_theorem).



> 
> **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 (slightly different version)? 
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 
>  $$\{e^\*\in\text{ext} \left( B\_{E^\*} \right): e^\*(e) = \|e\|\text{ for some }e\in B\}$$
>  norm-dense in $\text{ext}\left( B\_{E^\*} \right)?$ 
> 
> 
> 


If there is an affirmative answer to my question, may I have reference? 




---


*Notation:* $E^\*$ is the conjugate space. $B\_{E^\*}$ is the closed unit ball of $E^\*$. 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 $B\_{E^\*}.$



Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?
If $B$ is a bounded convex set that doesn't contain $0$, the Hahn-Banach separation theorem says there are $e^\* \in E^\*$ with $\| e^\*\| = 1$ and $\epsilon > 0$ with $e^\*(e) < -\epsilon$ for all $e \in B$; moreover, this is true (maybe with a smaller $\epsilon$) in a neighbourhood of $e^\*$. If $E^\*$ is strictly convex, so every point of its unit sphere is an extreme point, your set is **not** dense in $\text{ext}(B\_{E^\*})$.


