A set of optimal curvature-constrained closed trajectories for N unmanned aerial vehicle (UAV) mobile base stations (MBSs) supporting K ground terminals (GTs) is explored, which maximizes the minimum received signal strength (RSS) at the GT along its trajectory. It is proved that circular UAV maneuver turns out be optimal and the Euclidean N-center problem find the centers of the optimal circular trajectories. Numerical results demonstrate that optimized UAV trajectories improve the worst-case RSS by 100% compared to random UAV trajectories.