I know it's hard but anything written down could be good.
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.
Look at the "sticky" posts in the LaTeX help forum.