Subadditivity of the entropy and its relation to BrascampLieb type inequalities
Abstract
We prove a general duality result showing that a BrascampLieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of BrascampLieb 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 $\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical BrascampLieb 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 $\R^n$ in this paper, we discuss extensions to other settings as well.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.0870
 Bibcode:
 2007arXiv0710.0870C
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Probability;
 26D15;
 94A17
 EPrint:
 This is the revised version taking into account points made by the referee