This paper revisits the conventional convex lifting method for space partition in static topologies by adding a series of guarantees for obstacle avoidance in the framework of multi-agent navigation strategies. Further, the present work proposes a new methodology that adapts the convex lifting method in such a dynamic environment. To complete the developments towards navigation, the paper introduces a Model Predictive Control method for trajectory planning and navigation for an agent and presents examples of integrator-like multi-agent systems in a cluttered environment prioritizing the safety and control objectives of each agent.