Continuity of the Conic Hull

In a real Hilbert space $V$, the conic hull of $G \subseteq V$ is the set cone\,$(G)$ consisting of all nonnegative linear combinations of elements of $G$. Many optimization problems are sensitive to the changes in cone\,$(G)$ that result from changes in $G$ itself. Motivated by one such problem, we derive necessary and sufficient conditions for the continuity of the conic hull.