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Access-control strategies and rational pricing in code-division multiple-access (CDMA) systems need exact knowledge of the available transmission capacity and its limiting boundaries. We use the term "capacity region" to specify the set of user-transmission demands that can be supported at the desired quality of service (QoS). In this paper, we investigate the geometrical properties of the capacity region for a fixed number of users in a CDMA radio network under general QoS characteristics and general power constraints. It turns out that, under very mild assumptions, the capacity region is convex, and has an appealing monotonicity property. As a side result, we develop an elementary theory for characterizing the existence of solutions to systems of linear equations with nonnegative elements, analogous to Perron-Frobenius' theory, but bypassing irreducibility.