A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

Entropy (Basel). 2018 May 30;20(6):418. doi: 10.3390/e20060418.


Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality that unifies both the Brascamp-Lieb inequality and Barthe's inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.

Keywords: Brascamp-Lieb inequality; Gaussian optimality; functional-entropic duality; hypercontractivity; image size characterization; network information theory.