Thread: Advanced Geometry - Convex sets and Extreme points proof

Let K subset of R^n be a convex set. We call x1 element of K an extreme point of K if K/{x1} is convex too.

Prove that x1 element of K is an extreme point iff cx + (1-c)y = x1 for 0<c<1, c element of R (real numbers as above) implies x = y = x1.

any answer or help will be greatly appreciated.

I know it's hard but anything written down could be good.

Originally Posted by maxgunn555

Let K subset of R^n be a convex set. We call x1 element of K an extreme point of K if K/{x1} is convex too.

Prove that x1 element of K is an extreme point iff cx + (1-c)y = x1 for 0<c<1, c element of R (real numbers as above) implies x = y = x1.

any answer or help will be greatly appreciated.

For , write C(x) for the following condition:



Notice that, in that condition, if then . So the condition can be written in the modified (but equivalent) form



Now suppose that satisfies the condition . Then satisfies the definition of convexity. The reason is that if then (because X is convex), but (because of the modified form of the condition ). Therefore

The converse implication comes straight from the definition. If is convex, and , then Thus cannot be equal to . That shows that the modified form of condition holds.

thank you very much. i understand now. btw where do you get all those symbols and mathematical notation to make posts with?

Originally Posted by maxgunn555

where do you get all those symbols and mathematical notation to make posts with?

Look at the "sticky" posts in the LaTeX help forum.