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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Slicing inequalities for subspaces of $L_p$
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by Alexander Koldobsky PDF
Proc. Amer. Math. Soc. 144 (2016), 787-795 Request permission

Abstract:

We prove slicing inequalities for measures of the unit balls of subspaces of $L_p$, $-\infty <p<\infty .$ For example, for every $k\in \mathbb {N}$ there exists a constant $C(k)$ such that for every $n\in \mathbb {N},\ k<n$, every convex $k$-intersection body (unit ball of a normed subspace of $L_{-k})$ $L$ in $\mathbb {R}^n$ and every measure $\mu$ with non-negative even continuous density in $\mathbb {R}^n,$ \[ \mu (L)\ \le \ C(k) \max _{\xi \in S^{n-1}} \mu (L\cap \xi ^\bot )\ |L|^{1/n} \ ,\] where $\xi ^\bot$ is the central hyperplane in $\mathbb {R}^n$ perpendicular to $\xi ,$ and $|L|$ is the volume of $L.$ This and other results are in the spirit of the hyperplane problem of Bourgain. The proofs are based on stability inequalities for intersection bodies and estimates for the Banach-Mazur distance from the unit ball of a subspace of $L_p$ to the class of intersection bodies.
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Additional Information
  • Alexander Koldobsky
  • Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
  • MR Author ID: 104225
  • Email: koldobskiya@missouri.edu
  • Received by editor(s): July 20, 2014
  • Received by editor(s) in revised form: December 26, 2014
  • Published electronically: May 28, 2015
  • Additional Notes: This work was partially supported by the US National Science Foundation, grant DMS-1265155.
  • Communicated by: Thomas Schlumprecht
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 787-795
  • MSC (2010): Primary 52A20; Secondary 46B07
  • DOI: https://doi.org/10.1090/proc12708
  • MathSciNet review: 3430854