Transmit beamforming is a promising way to increase the downlink capacity of wireless networks. Since all users are coupled via their radiation patterns, the beamforming vectors must be optimized along with power control. It is necessary to balance the signal-to-interference levels according to individual QoS requirements. This problem leads back to the minimization of the infinity-norm of a certain vector and has first been studied by Gerlach and Paulraj in [1]. It has been assumed that the optimum solution can be obtained by minimizing the 1-norm instead, thereby leading to a new problem, which is generally easier to handle. The analytical and numerical results in this paper, however, indicate that this conjecture is generally not valid. We characterize the case where the 1-norm solution also solves the infinity-norm problem. In particular, it is shown that for the special case of a 2-user scenario, both optimization problems are indeed equivalent and a closed-form solution can be given. The analytical results provide new insights into the problem of coupled downlink beamforming and offer a useful approach to the design of efficient and reliable algorithms.