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Hypercontractive Measures, Talagrand’s Inequality, and Influences

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2050))

Abstract

We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.

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Acknowledgements

We thank F. Barthe and P. Cattiaux for their help with the bound (26), and R. Rossignol for pointing out to us the references [20, 21]. We also thank J. van den Berg and D. Kiss for pointing out that the techniques developed here cover the example of the complete graph and for letting us know about [14].

Author information

Authors and Affiliations

  1. Institut de Mathématiques, Université Pierre et Marie Curie (Paris 6 – Jussieu), 4 place Jussieu, 75005, Paris, France

    Dario Cordero-Erausquin

  2. Institut Universitaire de France, Paris, France

    Dario Cordero-Erausquin & Michel Ledoux

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

    Michel Ledoux

Authors
  1. Dario Cordero-Erausquin
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  2. Michel Ledoux
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Corresponding author

Correspondence to Dario Cordero-Erausquin .

Editor information

Editors and Affiliations

  1. Tel-Aviv, 69978, Israel

    Bo'az Klartag

  2. Technion Machine Learning Center, Technion, Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Shahar Mendelson

  3. Fac. Exact Sciences, School of Mathematical Sciences, University of Tel Aviv, Tel Aviv, 69978, Israel

    Vitali D. Milman

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Cordero-Erausquin, D., Ledoux, M. (2012). Hypercontractive Measures, Talagrand’s Inequality, and Influences. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_10

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