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Proceedings of the American Mathematical Society

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Slicing inequalities for subspaces of $L_p$


Author: Alexander Koldobsky
Journal: Proc. Amer. Math. Soc. 144 (2016), 787-795
MSC (2010): Primary 52A20; Secondary 46B07
DOI: https://doi.org/10.1090/proc12708
Published electronically: May 28, 2015
MathSciNet review: 3430854
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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
Article copyright: © Copyright 2015 American Mathematical Society