Let $V=\{v_k\}$ be a collection of vectors of $\Bbb{R}^n$, and define their cone to be the set of all their non-negative linear combinations: $$ C(V):=\Big\{ \sum_k a_k\,v_k; \; a_k\ge 0 \Big\}\;. $$
Now, is there a standard measure of width for this cone? One I would have in mind is the following: $$ W(C(V)) = \max \{ d(v_i,v_j); v_i,v_j\in V \}\;, $$ where we take as distance (for example) the angle: $$ d(v_i,v_j) = \arccos \dfrac{|v_i\cdot v_j|}{|v_i||v_j|}\;. $$
Does anything similar to this already exist? Maybe realized in a completely different way? Any reference would also be welcome.