I am reading a paper and there is such description as title. Why?

I have an example: $(0,1)$. This is a convex set but not closed, so I cannot find an extreme point. However if convex and compact,

I read some related problems:

  1. Exposed point of a compact convex set
    There must be at least one exposed point. But an extreme point is not necessary equal to an exposed point.
  2. Convex hull of extreme points
    A convex hull $P$ of finite points. Then $P$ is the convex hull of its extreme points.

It seems there is a requirement "finite points" to guarantee the topic?

  • $\begingroup$ Duplicate? math.stackexchange.com/questions/1384579/… $\endgroup$
    – A.Γ.
    Jul 2 '16 at 22:10
  • $\begingroup$ Yes, that is what I want. $\endgroup$
    – Denny
    Jul 2 '16 at 22:13
  • $\begingroup$ What is the setting here, are we talking about Euclidean space $\mathbb{R}^n$ or a normed space, or a locally convex topological space? $\endgroup$
    – user147263
    Jul 3 '16 at 2:53
  • $\begingroup$ Actually the paper is about positive semidefinite matrices with unity trace and rank one, which form extreme points $\endgroup$
    – Denny
    Jul 3 '16 at 21:33

In a finite dimensional space (which is the case here, according to a comment), the existence of an extreme point of a compact convex set $K$ is easy to prove. Take any point $x\in K$ at which the norm $\|x\|$ is maximized. If there is $y\ne 0$ such that $x\pm y \in K$, then $$ 2\|x\|^2 \ge \|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2+2\|y\|^2> 2\|x\|^2 $$ a contradiction.

  • $\begingroup$ What is this contradicting? That $x$ is not and extreme point? How do I know there is a $y$ such that $x \pm y \in K$? $\endgroup$
    Mar 2 '20 at 19:40

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.