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On the Distribution of the ψ2-Norm of Linear Functionals on Isotropic Convex Bodies

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Geometric Aspects of Functional Analysis

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

Abstract

It is known that every isotropic convex body K in \({\mathbb{R}}^{n}\) has a “subgaussian” direction with constant \(r\,=\,O(\sqrt{\log n})\). This follows from the upper bound \(\vert {\Psi }_{2}(K){\vert }^{1/n}\,\leq \,\frac{c\sqrt{\log n}} {\sqrt{n}} {L}_{K}\) for the volume of the body Ψ 2(K) with support function \({h}_{{\Psi }_{2}(K)}(\theta ) :{=\sup }_{2\leq q\leq n}\frac{\|\langle \cdot,{\theta \rangle \|}_{q}} {\sqrt{q}}\). The approach in all the related works does not provide estimates on the measure of directions satisfying a ψ2-estimate with a given constant r. We introduce the function \({\psi }_{K}(t) := \sigma (\{\theta \in {S}^{n-1} : {h}_{{\Psi }_{2}(K)}(\theta )\leq \mathit{ct}\sqrt{\log n}{L}_{K}\})\) and we discuss lower bounds for ψ K (t), \(t\geq 1\). Information on the distribution of the ψ2-norm of linear functionals is closely related to the problem of bounding from above the mean width of isotropic convex bodies.

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Acknowledgements

We would like to thank the referee for useful comments regarding the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Athens, Panepistimioupolis, 157 84, Athens, Greece

    Apostolos Giannopoulos & Petros Valettas

  2. Department of Mathematics, Texas A & M University, College Station, TX, 77843, USA

    Grigoris Paouris

Authors
  1. Apostolos Giannopoulos
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  2. Grigoris Paouris
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  3. Petros Valettas
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Correspondence to Apostolos Giannopoulos .

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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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Giannopoulos, A., Paouris, G., Valettas, P. (2012). On the Distribution of the ψ2-Norm of Linear Functionals on Isotropic Convex Bodies. 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_13

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