Strict Log-Convexity of the Minimum Power Vector

In a previous work, it was shown that the l¹-norm of the minimum power vector is log-convex if the signal-to-interference ratio (SIR) is establish log-convex function of the quality-of-service (QoS) value of interest. Assuming irreducibility of the gain matrix, the l¹-norm was further shown to be strictly log-convex in two special cases of SIR-QoS functions. In this paper, we extend this result in two directions: first, we drop the requirement on irreducibility of the gain matrix and, secondly, we consider any log-convex SIR-QoS relationship. Finally, we point out a connection of these results to strong convexity of a utility-based power control problem