This paper addresses the navigation problem in a multi-obstacle environment and makes use of convex lifting in trajectory planning problems with anti-collision constraints. The design problem is commonly stated in the literature in terms of a constrained optimization problem over a non-convex domain. In our framework, the convex lifting approach plays an instrumental role in the partitioning of the feasible space in accordance with the distribution of obstacles and in the subsequent generation of corridors in cluttered environments. We consider an adaptation of the generic MPC (Model Predictive Control) trajectory tracking problem, aiming to guarantee the feasibility and the convergence. Simulation results and proof of concepts illustrations prove the effectiveness of the proposed approach.