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

  • G. PaourisEmail author
Chapter
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 B2 n 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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Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CreteIraklionGreece

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