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2017 | OriginalPaper | Chapter

A Remark on Measures of Sections of $\boldsymbol{L}_{p}$-balls

Authors : Alexander Koldobsky, Alain Pajor

Published in: Geometric Aspects of Functional Analysis

Publisher: Springer International Publishing

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Abstract

We prove that there exists an absolute constant C so that

$$\displaystyle{ \mu (K)\ \leq \ C\sqrt{p}\max _{\xi \in S^{n-1}}\mu (K \cap \xi ^{\perp })\ \vert K\vert ^{1/n} }$$

for any p > 2, any $n \in \mathbb{N},$ any convex body K that is the unit ball of an n-dimensional subspace of L _p, and any measure μ with non-negative even continuous density in $\mathbb{R}^{n}.$ Here ξ ^⊥ is the central hyperplane perpendicular to a unit vector ξ ∈ S ⁿ⁻¹, and | K | stands for volume.

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previous chapter Super-Gaussian Directions of Random Vectors

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Title: A Remark on Measures of Sections of -balls
Authors: Alexander Koldobsky
Alain Pajor
Publisher: Springer International Publishing
Book: Geometric Aspects of Functional Analysis
Print ISBN: 978-3-319-45281-4

Electronic ISBN: 978-3-319-45282-1

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-45282-1_14

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A Remark on Measures of Sections of \(\boldsymbol{L}_{p}\)-balls

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