I've shown that the following result is valid for $V=\Bbb R^v$:
If $K\subset V$ is compact and convex and $V$ is a $v$-dimensional Banach space, then $K$ has at least one extreme points.
The proof I know uses the parallelogram law. Can we do something similar for general finite dimensional Banach spaces?
Obs.: The proof of Krein-Milman Theorem (for locally convex topological vector spaces) needs the existence of extreme points in general, and the proof of this uses Zorn's Lemma. I was wondering if there is a proof using something different, maybe similar to the parallelogram law...