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

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.

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

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Correspondence to Apostolos Giannopoulos .

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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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