How to prove that $C=\{x: Ax\le 0\}$ has at most one extreme point?

Question

Show that $C=\{x: Ax\le 0\},$ where $A$ is an $m\times n$ matrix, has at most one extreme point, the origin.

I tried to do the proof by contradiction but I don't know how to continue.

Attempt: (By contradiction)

Let's suppose $C$ has more than 1 extreme point, i.e. 2 extreme points, let's say $x,y\in C$ are such that

if $x=\lambda x_1+(1-\lambda)x_2,x_1,x_2\in C,\lambda\in(0,1),$ then $x=x_1=x_2$.

Similarly, $y=\lambda y_1+(1-\lambda)y_2,y_1,y_2\in C,\lambda\in(0,1),$ then $y=y_1=y_2$, by definition.

Also we have that $Ax\le0,Ay\le0$ and the same happens for $x_i$ and $y_i$.

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What can I do from here?

Could anyone help me, please?

Answer 1

Take $x\in C$ non zero. We will show that $x$ is not an extreme point.

Since $Ax\leq 0$ it implies that $A(2x)\leq 0$ therefore $2x \in C$. Also $0\in C$.

Now, notice that $$x= \frac{1}{2} \cdot 2x+\frac{1}{2} 0.$$

Hence, $x$ is not an extreme point.