Strict Convexity of the QoS Feasible Region for Log-Convex Interference Functions

We study the quality-of-service (QoS) feasible region of a multiuser system, under the assumption that the QoS is a bijective function of the signal-to-interference ratio (SIR). The inverse function is assumed to be log-convex (e.g. log-SIR). We derive a necessary and sufficient condition for strict convexity of the QoS region. This property holds for the class of log-convex interference functions, which include linear interference functions (resulting from single user receivers) and worst-case interference functions as special cases. Strict convexity is a desirable property, which ensures that optimization over the boundary of the region always leads to a unique global optimum. Moreover, we provide a necessary and sufficient condition for the strict convexity of a weighted cost/utility function, which is used in the context of resource allocation and scheduling.