We present a general framework of a resource theory based on the assumption of a) convexity and b) that the overlap of free states with maximally resourceful state is bounded. Using this structure, we derive bounds on the one-shot distillation rate for such a resource theory, thereby reproducing known bounds in coherence and entanglement. We use the free robustness and introduce a function Gmin related to overlap between states to express our bounds. To deal with resource theories where the free robustness in not finite we introduce the notion of imperfect free operations which we call ϵ-resource generating operations and generalize the free robustness toϵ-free robustness. We construct an ϵ-resource generating map that achieves pure state transformations and derive the conditions for such a transformation in terms of the ϵ-free robustness.