BIBO stability is a central concept for convolution systems, introduced in control theory by Callier, Desoer and Vidyasagar, in the seventies. It means that a bounded input leads to a bounded output, and is characterized by the fact that the kernel of the system is integrable. We generalize this result in the present paper, giving conditions for the output of a convolution system to evolve in a given poly-hedron, for any input evolving in another given convex polyhedron. The conditions are formulated in terms of integrals deduced from the kernel of the considered system and the given polyhedra. The condition is exact. It permits to construct exact inner and outer polyhedral approximations of the reachable set of a linear system. The result is compared to various known results, and illustrated on the example of a system with two delays.