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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The geometry of Euclidean convolution inequalities and entropy


Authors: Dario Cordero-Erausquin and Michel Ledoux
Journal: Proc. Amer. Math. Soc. 138 (2010), 2755-2769
MSC (2010): Primary 42A85, 52A40, 60E15; Secondary 60G15, 94A17
Published electronically: April 21, 2010
Previous version: Original version posted March 26, 2010
Corrected version: Current version corrects publisher's introduction of $=\mu_2$ in the first sentence of the fourth paragraph and the introduction of $_n$ at the end of the second line of the fifth paragraph.
MathSciNet review: 2644890
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Abstract: The goal of this paper is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $ \mathbb{R}^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.


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Additional Information

Dario Cordero-Erausquin
Affiliation: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252 Paris Cedex 05, France
Email: cordero@math.jussieu.fr

Michel Ledoux
Affiliation: Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse, France
Email: ledoux@math.univ-toulouse.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10304-9
PII: S 0002-9939(10)10304-9
Received by editor(s): July 16, 2009
Published electronically: April 21, 2010
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.