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

Area of Science:

• Information Theory
• Functional Analysis
• Probability Theory

Background:

• The Brascamp-Lieb inequality is a fundamental result in geometric analysis and information theory.
• Barthe's inequality provides a reverse form, crucial for understanding the behavior of certain transformations.
• Network information theory problems often involve characterizing channel capacities and information flow.

Purpose of the Study:

• To introduce a novel functional inequality unifying the Brascamp-Lieb and Barthe inequalities.
• To establish an equivalent entropic formulation of this unified inequality.
• To demonstrate Gaussian optimality for Gaussian reference measures.

Main Methods:

• Development of a new functional inequality inspired by network information theory.
• Proof of the entropic formulation using Legendre-Fenchel duality theory on Polish spaces.
• Application of a 'doubling trick' technique for proving Gaussian optimality.

Main Results:

• A unified functional inequality is introduced, generalizing existing inequalities.
• An equivalent entropic formulation is proven for Polish spaces.
• Gaussian optimality is established for Gaussian reference measures.

Conclusions:

• The unified inequality provides a broader framework for studying related problems.
• The entropic formulation and proof techniques offer new tools for analysis.
• The findings contribute to a deeper understanding of inequalities in information theory and analysis.