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The high Q-factor (low bandwidth) and low efficiency make the design of small antennas challenging. Here, convex optimization is used to determine current distributions that provide upper bounds on the antenna performance. Optimization formulations for maximal gain Q-factor quotient, minimal Q-factor for superdirectivity, and minimum Q for given far-fields are presented. The effects of antennas embedded in structures are also discussed. The results are illustrated for planar geometries.