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

Question

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?

Answer 1

If $(x,y)$ is in $S$ then so is $(x+1,y-1)$ and $(x-1,y+1)$. What is $.5(x+1,y-1) + .5(x-1,y+1)$?

Answer 2

Big hint: If $(x,y)\in S$ then also $(x+z,y-z)\in S$ for any real number $z$.