Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

By Eric A. Carlen, Dario Cordero-Erausquin. 2007

115 citations recorded by the source corpus.

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $${\mathbb {R}^n}$$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schrödinger operators. Though we focus mainly on the case of random variables in $${\mathbb {R}^n}$$ in this paper, we discuss extensions to other settings as well.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/5ac2ec32b56e522e68de4ef5ae3bd077cc34cc7e

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In Applied Mathematics

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Correlation and Brascamp–Lieb Inequalities for Markov Semigroups (2009, methodology): “(20) A recent work by Carlen and Cordero-Erausquin [10] shows that there is a full equivalence: Proposition 23.”
Log-Concavity and Strong Log-Concavity: a review (2014, result): “More recently still, further results have been obtained by: Carlen, Lieb and Loss (2004), Carlen and Cordero-Erausquin (2009), and Cordero-Erausquin and Ledoux (2010).”
Cone-volume measure of general centered convex bodies (2016, unknown): “Cordero-Erausquin [14], and J.”
Representation formula for the entropy and functional inequalities (2010, background): “As observed by Carlen and Cordero [10], this super-additivity property of the relative entropy is equivalent to the following Brascamp-Lieb inequality.”
Affine images of isotropic measures (2015, background): “[5] F. Barthe & D. Cordero-Erausquin, Invariances in variance estimates, Proc.”