Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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,$

$\displaystyle \mu (L)\ \le \ C(k) \max _{\xi \in S^{n-1}} \mu (L\cap \xi ^\bot )\ \vert L\vert^{1/n} \ ,$

where $ \xi ^\bot $ is the central hyperplane in $ \mathbb{R}^n$ perpendicular to $ \xi ,$ and $ \vert L\vert$ 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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52A20, 46B07

Retrieve articles in all journals with MSC (2010): 52A20, 46B07


Additional Information

Alexander Koldobsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: koldobskiya@missouri.edu

DOI: https://doi.org/10.1090/proc12708
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