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

  • Apostolos Giannopoulos
  • Grigoris Paouris
  • Petros Valettas
Chapter
Part of the Lecture Notes in Mathematics book series (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.

Keywords

Convex Body Absolute Constant Logarithmic Term Invariant Probability Measure Symmetric Convex Body 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Apostolos Giannopoulos
    • 1
  • Grigoris Paouris
    • 2
  • Petros Valettas
    • 1
  1. 1.Department of MathematicsUniversity of AthensAthensGreece
  2. 2.Department of MathematicsTexas A & M UniversityCollege StationUSA

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