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Ja n 20 08 Subadditivity of the Entropy and its Relation to Brascamp – Lieb Type Inequalities
| Content Provider | Semantic Scholar |
|---|---|
| Author | Erausquin, Dario Cordero – |
| Copyright Year | 2008 |
| Abstract | 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 Rn, 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 Rn in this paper, we discuss extensions to other settings as well. Mathematics Subject Classification Numbers: 26D15, 94A17 Work partially supported by U.S. National Science Foundation grant DMS 06-00037. c © 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. 1 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0710.0870v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Convolution Fisher information Generalization (Psychology) Leg length inequality in children Leggett–Garg inequality Mathematics Subject Classification Schrödinger Social inequality |
| Content Type | Text |
| Resource Type | Article |
