Strict convexity of the feasible log-SIR region

The feasible log-SIR region is defined as a set of all signal-to-interference ratios (SIR) expressed in logarithmic scale that can be supported in a wireless network by means of power control and with all users being active concurrently. Recently, the feasible log-SIR region was shown to be a convex set, which is a key ingredient in the development of some power control strategies for wireless systems. In this paper, under the assumption of a noiseless channel, we strengthen these results by proving a necessary and sufficient condition for the feasible log-SIR region to be a strictly convex set. The strict convexity property is of interest since it is closely related to the problem of the existence and uniqueness of a so-called log-SIR fair power vector.