Why a convex cone cannot have more than one extreme point?

Question

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

Answer 1

Geometric Intuition of Extreme Point: According to your definition of extreme point, geometric intuition of an extreme point in a convex are all corners. For example, in a triangle you extreme points are just the corners. However, in some other convex sets like disk (filled circle), extreme points are all the points on the border.

Geometric Intuition of Convex Cone: Look at this link for a geometric description of a convex cone. http://en.wikipedia.org/wiki/Convex_cone

Observation: According to these the two statements above, you can see that only the origin can be an extreme point for a convex cone. So, if the convex cone includes the origin it has only one extreme point, and if it doesn't it has no extreme points.

Answer 2

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.)