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


  • $\begingroup$ @CaveJohnson So how could you define "extreme point" in a non-convex set? $\endgroup$ – Filburt Aug 4 '17 at 4:36

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.

  • 1
    $\begingroup$ How do you know that $K$ can be inscribed in a sphere? $\endgroup$ – Robert Israel Aug 4 '17 at 6:34

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).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.