# How to prove that $S=\{(x,y) \in \mathbb{R}^2 : x+y \leq 2 \}$ has no extreme points?

Let

$$S := \left\{ (x,y) \in \mathbb{R}^2 : x + y \leq 2 \right\}$$

Definition: A point $$x \in S$$ is extreme if it cannot be written as a convex combination of other elements of $$S$$.

I started trying to show that $$(1,1)$$ was not an extreme point, but I couldn't get anywhere to make a general case, or what can I use?

• Are you sure $(x,y) \in \mathbb{R}^2$? Did you mean $(x,y) \in \mathbb{R}_{\geq 0}^2$? – Rodrigo de Azevedo Sep 8 '19 at 6:47

If $$(x,y)$$ is in $$S$$ then so is $$(x+1,y-1)$$ and $$(x-1,y+1)$$. What is $$.5(x+1,y-1) + .5(x-1,y+1)$$?
Big hint: If $$(x,y)\in S$$ then also $$(x+z,y-z)\in S$$ for any real number $$z$$.