Given $U$ an uncertainty set, I don't see the point in saying that
"For most models of robust optimization, it is easy to prove that changing from $U$ to its convex hull does not change the problem, as the worst case in the inner maximization problem will be attained in an extreme point anyway." (page 3, 2.2 ).
Or similarly,
"The uncertainty set $U$ can be replaced by its convex hull $\operatorname{conv}(U)$, i.e., the smallest convex set that includes $U$, because testing the feasibility of a solution with respect to $U$ is equivalent to taking the supremum of the left hand side of a constraint over $U$, which yields the same optimal objective value if the maximization is $\operatorname{conv}(U)$." (page 3, E.3)
If the problem described here is already convex, what is the sense to replacing $U$ with its hull?