Let $E\subseteq \mathbb{R}^2$, the diameter of $E$ is defined as $diam(E):=\sup\limits_{x,y\in E}d(x,y)$ where $d$ is standard euclidean distance.
If $P$ is a $n$-polygon in the plane with vertices $V_i$ ($i=1,...,n$), is true that $diam(P)=\max\limits_{i,j} d(V_i,V_j)$ ?
I ask please help for to proof (assumed polygon convex?) or to find a counterexample.
Ps. $diam(P)=\max\limits_{x,y\in E}d(x,y)$ because $P$ is compact and we can suppose that $diam(P)=d(x^*,y^*)$ with $x^*,y^*\in\partial P$