Diameter of a finite convex hull ($\mathrm{diam}(\mathrm{conv}\{v_1,\dots ,v_m\})={\max }_{i,j}\Vert v_i-v_j\Vert$)

In a normed vector space, the diameter of the convex hull of finitely many points equals their greatest pairwise distance. For $x={\sum }_i{\lambda }_i{v}_i$ and $y={\sum }_j{\mu }_j{v}_j$, with nonnegative coefficients each summing to one,The triangle inequality gives $\Vert x-y\Vert \le {\sum }_{i,j}{\lambda }_i{\mu }_j\Vert v_i-v_j\Vert \le {\max }_{i,j}\Vert v_i-v_j\Vert$. The reverse inequality holds because all the vertices belong to the convex hull. Thus a continuum of distances inside a convex polygon can be controlled by finitely many vertex pairs.