Diameter of finite set of points is equal to diameter of its convex hull Let $M\subset \mathbb{R}^2$ be a finite set of points, $\operatorname{C}(M)$ the convex hull of M and 
$$\operatorname{diam}(M) = \sup_{x,y\in M}\|x-y\|_2$$
be the diameter of $M$
What I want to show now is, that it holds
$$\operatorname{diam}(M) = \operatorname{diam}(\operatorname{C}(M))$$
Because $$M\subseteq\operatorname{C}(M)$$ we obtain $$\operatorname{diam}(M) \le\operatorname{diam}(\operatorname{C}(M))$$ but how to proof that $$\operatorname{diam}(M) \ge \operatorname{diam}(\operatorname{C}(M))$$
I suppose it should be possible to construct a contradiction assuming $\operatorname{diam}(M) <\operatorname{diam}(\operatorname{C}(M))$ but i do not see how at this moment. 

Hint:
Prove this for a triangle and then use the fact that for every point of $C(M)$ there is a triangle that contains it, there are many ways to go from there.
I hope this helps ;-)
