# Advanced Geometry - Convex sets and Extreme points proof

• November 28th 2011, 09:23 AM
maxgunn555
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.
• November 30th 2011, 05:38 PM
maxgunn555
Re: Advanced Geometry - Convex sets and Extreme points proof
I know it's hard but anything written down could be good.
• December 1st 2011, 05:24 AM
Opalg
Re: Advanced Geometry - Convex sets and Extreme points proof
Quote:

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 $x\in X$, write C(x) for the following condition:

$C(x)\qquad \text{If } x = cy+(1-c)z \text{ (where }y,z\in X, 0

Notice that, in that condition, if $y\ne x$ then $z\ne x$. So the condition can be written in the modified (but equivalent) form

$C(x)\qquad \text{If }y,z\in X\setminus\{x\} \text{ and } 0

Now suppose that $x_1$ satisfies the condition $C(x_1)$. Then $X\setminus\{x_1\}$ satisfies the definition of convexity. The reason is that if $y,z\in X\setminus\{x_1\}$ then $cy+(1-c)z\in X$ (because X is convex), but $cy+(1-c)z\ne x_1$ (because of the modified form of the condition $C(x_1)$). Therefore $cy+(1-c)z\in X\setminus \{x_1\}.$

The converse implication comes straight from the definition. If $X\setminus \{x_1\}$ is convex, and $y,z\in X\setminus \{x_1\}$, then $cy+(1-c)z\in X\setminus \{x_1\}.$ Thus $cy+(1-c)z$ cannot be equal to $x_1$. That shows that the modified form of condition $C(x_1)$ holds.
• December 1st 2011, 04:22 PM
maxgunn555
Re: Advanced Geometry - Convex sets and Extreme points proof
thank you very much. i understand now. btw where do you get all those symbols and mathematical notation to make posts with?
• December 2nd 2011, 04:14 AM
Opalg
Re: Advanced Geometry - Convex sets and Extreme points proof
Quote:

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.