Proving x* is an extreme point using S\{x*}

[ploppers]

Homework Statement

Let x* be an element of a convex set S. Show that x* is an extreme point of S if and only if the set S\{x*} is a convex set.

Homework Equations

(1-λ)x1 + λx2 exists in the convex set

The Attempt at a Solution

I'm not too sure what S\{x*}, I asssumed it was the same as S/{x*} which is S over {x*}
I have is S?{x*} is a convex set then
λ(K/x*) + (1-λ)(P/x*) is a convex set were K and P are in the convex set S.
[λ(k) + (1-λ)(P)]/x* is in S/{x*}, but I can't see how it must me an extreme point

 

[CompuChip]

Usually \ is read as "minus":
$$S \setminus \{ x^* \} = \{ s \in S \mid s \neq x^* \}$$

I suppose the statement is intuitive: you can only keep drawing straight lines between points, if the point you take out is on an "edge".

 

[ploppers]

Ahh thanks, I should have looked up the notation haha!