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Proceedings of the American Mathematical Society

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Estimates for moments of general measures on convex bodies

Authors: Sergey Bobkov, Bo’az Klartag and Alexander Koldobsky
Journal: Proc. Amer. Math. Soc. 146 (2018), 4879-4888
MSC (2010): Primary 52A20; Secondary 46B07
Published electronically: July 23, 2018
MathSciNet review: 3856154
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Abstract: For $ p\ge 1$, $ n\in \mathbb{N}$, and an origin-symmetric convex body $ K$ in $ \mathbb{R}^n,$ let

$\displaystyle d_{\rm {ovr}}(K,L_p^n) = \inf \left \{ \Big (\frac {\vert D\vert}{\vert K\vert}\Big )^{1/n}: K \subseteq D,\ D\in L_p^n \right \}$    

be the outer volume ratio distance from $ K$ to the class $ L_p^n$ of the unit balls of $ n$-dimensional subspaces of $ L_p.$ We prove that there exists an absolute constant $ c>0$ such that

$\displaystyle \frac {c\sqrt {n}}{\sqrt {p\log \log n}}\le \sup _K d_{\rm {ovr}}(K,L_p^n)\le \sqrt {n}.$    

This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant $ C>0$ so that for any $ p\ge 1,$ any $ n\in \mathbb{N}$, any compact set $ K \subseteq \mathbb{R}^n$ of positive volume, and any Borel measurable function $ f\ge 0$ on $ K$,

$\displaystyle \int _K f(x)\,dx \, \le \, C \sqrt {p}\ d_{\rm ovr}(K,L_p^n)\ \vert K\vert^{1/n} \sup _{H} \int _{K\cap H} f(x)\,dx,$    

where the supremum is taken over all affine hyperplanes $ H$ in $ \mathbb{R}^n$. Combining the above display with a recent counterexample for the slicing problem with arbitrary measures from the work of the second and third authors [J. Funct. Anal. 274 (2018), pp. 2089-2112], we get the lower estimate from the first display.

In turn, the second inequality follows from an estimate for the $ p$-th absolute moments of the function $ f$

$\displaystyle \min _{\xi \in S^{n-1}} \int _K \vert(x,\xi )\vert^p f(x)\ dx \, \le \, (Cp)^{p/2}\, d^p_{\rm {ovr}}(K,L_p^n)\ \vert K\vert^{p/n} \int _K f(x)\,dx.$    

Finally, we prove a result of the Busemann-Petty type for these moments.

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Additional Information

Sergey Bobkov
Affiliation: Department of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, Minnesota 55455

Bo’az Klartag
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 Israel – and – School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978

Alexander Koldobsky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Received by editor(s): December 16, 2017
Received by editor(s) in revised form: February 1, 2018
Published electronically: July 23, 2018
Additional Notes: This material is based upon work supported by the U. S. National Science Foundation under Grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The first- and third-named authors were supported in part by the NSF Grants DMS-1612961 and DMS-1700036. The second-named author was supported in part by a European Research Council (ERC) grant.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2018 American Mathematical Society