I define an extreme point as a point $x$ (in some set $S$) which cannot be defined as a convex combination of two distinct points $x_1$ and $x_2$ (both in $S$). I.e., if $x=\lambda x_1+(1-\lambda) x_2$ for distinct $x_1$ and $x_2$, and $\lambda\in [0,1]$, then $x=x_1=x_2$.
I'm not able to extend this and show why a convex cone cannot have more than one extreme point. Here, a convex cone is a (i) cone $C$, i.e. for all $x\in C$, $\lambda x\in C$ for all $\lambda\geq 0$, which (ii) is convex.
Can someone give me more geometric intuition behind this concept?
This was asked in my exam, and I was not convinced with my prof's explanation.
Thank you
Best Answer
Let $C$ be a cone and let $x \in C$. Suppose $x \neq 0$.
Then $x$ is a convex combination of the points $y_1 = \frac12 x$ and $y_2 = \frac32 x$, both of which belong to $C$.
Explicitly, $x = \frac12 y_1 + \frac12 y_2$.
This shows that $x$ is not an extreme point of $C$. It follows that the origin is the only possible extreme point of $C$.
(We don't need the assumption that $C$ is convex.)