Extreme point of a convex set

Let S be a convex set in $mathbb{R}^n$. A vector $x in S$ is said to be a extreme point of S if $x= lambda x_1+left ( 1-lambda ight )x_2$ with $x_1, x_2 in S$ and $lambda inleft ( 0, 1 ight )Rightarrow x=x_1=x_2$.

Example

Step 1 − $S=left { left ( x_1,x_2 ight ) in mathbb{R}^2:x_{1}^{2}+x_{2}^{2}leq 1 ight }$

Extreme point, $E=left { left ( x_1, x_2 ight )in mathbb{R}^2:x_{1}^{2}+x_{2}^{2}= 1 ight }$

Step 2 − $S=left { left ( x_1,x_2 ight )in mathbb{R}^2:x_1+x_2< 2, -x_1+2x_2leq 2, x_1,x_2geq 0 ight }$

Extreme point, $E=left { left ( 0, 0 ight), left ( 2, 0 ight), left ( 0, 1 ight), left ( frac{2}{3}, frac{4}{3} ight) ight }$

Step 3 − S is the polytope made by the points $left { left ( 0,0 ight ), left ( 1,1 ight ), left ( 1,3 ight ), left ( -2,4 ight ),left ( 0,2 ight ) ight }$

Extreme point, $E=left { left ( 0,0 ight ), left ( 1,1 ight ),left ( 1,3 ight ),left ( -2,4 ight ) ight }$

Remarks

Any point of the convex set S, can be represented as a convex combination of its extreme points.

It is only true for closed and bounded sets in $mathbb{R}^n$.

It may not be true for unbounded sets.

k extreme points

A point in a convex set is called k extreme if and only if it is the interior point of a k-dimensional convex set within S, and it is not an interior point of a (k+1)- dimensional convex set within S. Basically, for a convex set S, k extreme points make k-dimensional open faces.