I find the definition a bit ambiguous (perhaps I am missing something).
The definition goes as follows:
A point $\textbf{x}$ in a convex set $S$ is said to be an extreme point of $S$ if for no distinct $\textbf{x}_1, \textbf{x}_2\in S$, $\textbf{x} = \lambda \textbf{x}_1 +(1-\lambda)\textbf{x}_2, \lambda\in (0,1)$
Now, what if the set $S$ is open? How does the definition hold up?.
My textbook says another version of it:
A point $\textbf{x}$ is said to be an extreme point of a convex set $S$ if for no distinct $\textbf{x}_1, \textbf{x}_2\in S$, $\textbf{x} = \lambda \textbf{x}_1 +(1-\lambda)\textbf{x}_2, \lambda\in (0,1)$
This seems even more ridiculous (perhaps I am mistaken, again). Because according to this definition, any point lying outside of $S$ is entitled to be an extreme point (i.e. doesn't even need to be a boundary point).
Can some explain what am I missing?
What I have been taught, is a more positively stated version, basically equivalent to your first: $p \in S$ is an extreme point of $S$ when
$$\forall x,y \in S: \forall t \in [0,1]: (tx + (1-t)y = p) \implies t\in \{0,1\}$$