We characterize a family of parametrized robust control invariant sets for linear discrete time systems subject to additive but bounded state disturbances. The existence of a member of the introduced family of parametrized robust control invariant sets can be verified by solving a tractable convex optimization problem in the linear convex case, which reduces to the standard linear or convex quadratic programme in the linear polytopic case. The developed method can also be utilized to detect and obtain an implicit representation of local control Lyapunov functions in the linear convex case from the solution of a single and tractable convex optimization problem. The offered method permits for the computation of polytopic robust control invariant sets and local control Lyapunov functions of indirectly controlled and limited complexity in the linear polytopic case.