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Remarks on Gaussian Noise Stability, Brascamp-Lieb and Slepian Inequalities

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Book cover Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaussian measure.

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Acknowledgements

This note grew up out of discussions and exchanges with J. Neeman and E. Mossel around their works [28, 30] and [18]. In particular, Sect. 2, the first part of Sects. 3 and 5 are directly following their contributions. I sincerely thank them for their interest and comments. I also thank K. Oleszkiewicz for exchanges on ρ-concave functions, R. Bouyrie for several comments and corrections, F. Barthe for pointing out the reference [17], and the referee for helpful comments in improving the exposition.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université de Toulouse, 31062, Toulouse, France

    Michel Ledoux

  2. Institut Universitaire de France, Paris, France

    Michel Ledoux

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  1. Michel Ledoux
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Correspondence to Michel Ledoux .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Ledoux, M. (2014). Remarks on Gaussian Noise Stability, Brascamp-Lieb and Slepian Inequalities. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_20

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