  The extremal inequality approach plays a key role in network information
theory problems. In this paper, we propose a novel monotone path construction
in product probability space. The optimality of Gaussian distribution is then
established by standard perturbation arguments. The proofs of Liu-Viswanath
extremal and vector Generalization of Costa's entropy power inequality are
illustrated into the unified framework. As applications, capacity region of the
multiple-input multiple-output (MIMO) Gaussian broadcast channel and
rate-distortion-equivocation function of the vector Gaussian secure source
coding are revisited through our proposed extremal inequality approach.
