I consider any set $A$ in $\mathbb{R}^n$. Note that $A$ is not necessary to be a convex set. I define the extreme point as follows: Given any set $C$, a point $a\in C$ is an extreme point of the set $C$ if we cannot find any two different points $a_0\in C,a_1\in C$ and $a_0\neq a_1$ such that $a=\lambda a_0 + (1-\lambda) a_1$, for any $\lambda\in(0,1)$

My question is: whether or not, the set of the extreme points of $A$ is the same as the set of the extreme points of $conv(A)$, where $conv(A)$ means the convex hull of set $A$.

I guess the two sets of extreme points above would be the same set since the operator $conv()$ should not generate or remove any extreme points of $A$. If the two sets are different, that means there exists a point in $conv(A)$ such that the point is the extreme point of $conv(A)$ but not the extreme point of $A$. I think this leads to a contradiction but I cannot vigorously argue it.



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