extreme point of a convex set is also a bounday point?

Question

Could anyone help me to show: if C is a convex set and x is an extreme point of C then it is also a boundary point of C. Thank you.

Answer 1

Let's say $x\in C$ is an extreme point. Suppose $x\notin \partial C=\bar C -\text{int} C$, then $x\in\text{int} C$. So $\exists r>0$ such that the closed ball $B(x,r)\subset C$, but then $x$ would be the midpoint of any antipodal pair on the surface of $B(x,r)$.

Answer 2

Suppose $x$ is not a boundary point of $C$.

Then there exists a $r \gt 0$, such that $B(x,r) \cap C = \emptyset$ or $B(x,r) \cap C^c=\emptyset$.

Since $x \in C$ ($\because x$ is an extreme point of $C$), $B(x,r) \cap C^c = \emptyset$. But this means that $B(x,r) \subset C$.

Since $C$ is convex, we can find two points $y$ and $z$ in $B(x,r)$ such that $ty+(1-t)z =x$ for some $t \in (0,1)$. This implies that $x$ is not an extreme point, a contradiction.