= Diameter of a finite convex hull
{title2=$\operatorname{diam}(\operatorname{conv}\{v_1,\ldots,v_m\})=\max_{i,j}\|v_i-v_j\|$}
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 $y=\sum_j\mu_jv_j$, with nonnegative coefficients each summing to one,
$$
x-y=\sum_{i,j}\lambda_i\mu_j(v_i-v_j).
$$
The <triangle inequality> gives $\|x-y\|\leq\sum_{i,j}\lambda_i\mu_j\|v_i-v_j\|\leq\max_{i,j}\|v_i-v_j\|$. 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.
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