Let $K$ be a convex set in topological vector space and $a$ an interior point of $K$. Can it be an extreme point of $K$?
Suppose $a$ is an extreme point of $K$. Then $a=(1-t)x + ty$ for $x\neq y \in K$ and $t\in [0,1]$. Then $t=1$ or $t=0$. But if that happened every point of $K$ is an extreme point. So interior of $K$ is empty. But $a$ belongs to interior of $K$. Hence the contradiction. So, $a$ is not an extreme point.
Is my logic correct?
Here is a valid proof. Let $a$ be an interior point of $K$ and $x$ be any non-zero vector. By the definition of a topolgical vector space $a+\frac 1 n x$ and $a-\frac 1 n x$ tends to $a$ as $n \to \infty$. Hence, these points belong to $K$ for $n$ sufficiently large. Since $a=\frac {(a+\frac 1 n x)+(a-\frac 1 n x)} 2$ it follows that $a$ is not an extreme point.