On the Existence and Uniqueness of Proportionally Fair Power Allocation

The geometry of the feasible log-SIR region has attracted a great deal of attention. In particular, it was shown that the feasible log-SIR region is a convex set, which is a key ingredient in the development of some power control strategies for wireless systems. In this paper, we strengthen this result by proving a necessary and sufficient condition for strict convexity of the feasible log-SIR region. The result sheds light on the existence and uniqueness of proportionally fair power vectors.