An Extremal Inequality With Application to Gray-Wyner System

This paper considers a new extremal inequality and proves that a Gaussian distribution is one of the solutions to hold this extremal inequality. The method of factorization is applied in the proof. We also exploit the chain rule in two separate ways to establish the subadditivity of factorization. As an application, the converse part of the rate-distortion problem of the quadratic Gaussian Gray-Wyner system with single-letter characterization is solved using this new extremal entropy inequality.