Given a convex set C argue that a point x elementof C is ext

Given a convex set C, argue that a point x elementof C is extreme if and if the set C\\{x} is convex. Here C\\{x} only denotes the set C after removing the point x.

Solution

Firstly, let xC be an extreme point. Then, by definition,

u,v C, uv, such that x = (1-p)u+pv ; p(0,1)                           - (1)

We need to show that C\\{x} is a convex set.

Let m,n C\\{x}. Then, m,nx.

Also, as C\\{x}C, m,nC.

=> (1-p)m+pn C, since C is convex.

=> (1-p)m+pn C\\{x}, by (1). [ Reason: If (1-p)m+pn=x, then it would contradict the fact that x is an extreme point.]

Therefore, C\\{x} is a convex set.

Conversely, let C\\{x} be a convex set.

We need to show that x is an extreme point. Let,on the contrary, x not be an extreme point.

Then, since C is convex,

m,nC such that x= (1-p)m+pn; p(0,1)                -(2)

Note that m,n C\\{x}, which is convex.

=> (1-p)m+pn C\\{x} ; p(0,1)

=> x C\\{x}, by (2), which is a contradiction.

Thus, x is an extreme point.

Hence proved.