A forward-reverse Brascamp-Lieb inequality: Entropic duality and Gaussian optimality

Jingbo Liu, Thomas A. Courtade, Paul W. Cuff, Sergio Verdú

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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.

Original languageEnglish (US)
Article number418
Issue number6
StatePublished - Jun 1 2018

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Electrical and Electronic Engineering
  • General Physics and Astronomy
  • Mathematical Physics
  • Physics and Astronomy (miscellaneous)


  • Brascamp-Lieb inequality
  • Functional-entropic duality
  • Gaussian optimality
  • Hypercontractivity
  • Image size characterization
  • Network information theory


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