Is there a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point?

Question

Can someone give me a simple proof that every compact convex $K\subset \Bbb R^n$ has at least one extreme point?

I'd found this one but couln't understand it: Convex compact set must have extreme points

Thanks!

Answer 1

Consider a sphere $S $ of radius $r$ where $K$ is inscribed in $S$.

Here inscribed implies that $K$ is in $r$-ball $B$ with a boundary $S$ and there is a point $x\in K$ s.t. $x$ is in $S$.

Then $x$ is an extreme point of $K$. If not, there is $p,\ q\neq p \in K$ s.t. $x$ is an interior point of a segment $[pq]$.

Note that $B$ can not contain a segment $[pq]$ so that $[pq]$ is not in $K$. It is a contradiction.

Answer 2

Take a point of $K$ that maximizes Euclidean distance from the origin. This exists since a continuous function on a compact set attains its supremum, and it is easily seen to be an extreme point (you may want to use the Parallelogram Law for this).