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Accurate power control is an essential requirement in the design of cellular code-division multiple-access (CDMA) systems. In this paper, we contribute three main themes to the power control problem. First, we derive an efficient algorithm for computing minimal power levels for large-scale networks within seconds. Nice and intuitive conditions for the existence of feasible power solutions follow from this approach. Second, we define the capacity region of a network by the set of effective spreading gains, or data rates, respectively, which can be supplied by the network. This is achieved by bounding the spectral radius of a certain matrix containing system parameters and mutual transmission gain information. It is shown that the capacity region is a convex set. Finally, we reveal an interesting duality between the uplink and downlink capacity region. In a clear-cut analytical way, it substantiates the fact that the uplink is the more restricting factor in cellular radio networks. The same methods carry over to certain models of soft handover. In the case that the channel gains are subject to log-normal shadowing, we introduce the concept of level-α capacity regions. Despite the complicated structure, it can still be shown that this set is sandwiched by two convex sets coming arbitrarily close as variance decreases.