On the convexity of feasible QoS regions

The feasible quality-of-service (QoS) region is the set of all QoS vectors that can be provided to the users by means of power control, with interference treated as noise. In an interference-limited scenario, this set is determined by the Perron root of some QoS-dependent nonnegative matrix. In a previous work, we showed that if the signal-to-interference ratio (SIR) is a log-convex function of the QoS, then the Perron root is a log-convex function. This implies convexity of the feasible QoS region. In this correspondence, we prove that the log-convexity property is also necessary for the Perron root to be convex for any choice of the (path) gain matrix. Interestingly, a significantly less restrictive property is sufficient when the gain matrix is confined to, be symmetric positive semidefinite.