Strong chip for infinite system of closed convex sets in normed linear spaces

For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior-point conditions together with appropriate continuity of the associated set-valued function on the (topologized) index set I. In the special case when I is finite and X is finite dimensional, one of these results reduces to a classical result of Rockafellar.