I need to show that given a convex set $K \subset \mathbb{R}^n$ and a point $\overline{x} \in K$ the following holds:
If $\overline{x}$ is a unique solution to the $LP$
$\min_{x \in K} \ c^Tx$
then $\overline{x}$ is an extremal point.
We defined an extremal point $a$ as a point in $K$ such that every convex combination in $K$ with $a = \lambda*b + (1-\lambda)*c$ has the property $\lambda = 1$ or $(1-\lambda) = 1$ $\forall \ a,b \in K \ with \ a \ne b$).
I tried doing this with a proof by contradiction, i.e.: assuming there was a $\lambda \in (0,1)$ such that $\overline{x} = \lambda*b + (1-\lambda)*c$ for some $b,c \in K$, but I don't see how I could go on with this approach.
Could you give me any hints?