The geometry of Euclidean convolution inequalities and entropy

Cordero-Erausquin, Dario; Ledoux, Michel

doi:10.1090/S0002-9939-10-10304-9

The geometry of Euclidean convolution inequalities and entropy
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by Dario Cordero-Erausquin and Michel Ledoux PDF

Proc. Amer. Math. Soc. 138 (2010), 2755-2769 Request permission

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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