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\(\Psi_2\)-Estimates for Linear Functionals on Zonoids

Part of the Lecture Notes in Mathematics book series (LNM,volume 1807)

Abstract

Let K be a convex body in \({\mathbb R}^n\) with centre of mass at the origin and volume |K| = 1. We prove that if \(K\subseteq\alpha\sqrt{n}B_2^n\) where B2n is the Euclidean unit ball, then there exists \(\theta\in S^{n-1}\) such that

$$\|\langle \cdot ,\theta\rangle \|_{L_{\psi_{2}}(K)}\leq c\alpha \|\langle \cdot ,\theta\rangle\|_{L_1(K)}, \qquad (*)$$

where c > 0 is an absolute constant. In other words, “every body with small diameter has \(\psi_2\)-directions”. This criterion applies to the class of zonoids. In the opposite direction, we show that if an isotropic convex body K of volume 1 satisfies (*) for every direction \(\theta\in S^{n-1}\), then \(K\subseteq C\alpha^2\sqrt{n}\log nB_2^n\), where C > 0 is an absolute constant.

Mathematics Subject Classification (2000):

  • 46-06
  • 46B07
  • 52-06 60-06

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Correspondence to G. Paouris .

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© 2003 Springer-Verlag Berlin/Heidelberg

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Paouris, G. (2003). \(\Psi_2\)-Estimates for Linear Functionals on Zonoids. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36428-3_17

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  • DOI: https://doi.org/10.1007/978-3-540-36428-3_17

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00485-1

  • Online ISBN: 978-3-540-36428-3

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