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Proceedings of the American Mathematical Society

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Remarks on an inequality of Rogers and Shephard


Authors: Apostolos Giannopoulos, Eleftherios Markessinis and Antonis Tsolomitis
Journal: Proc. Amer. Math. Soc. 144 (2016), 763-773
MSC (2010): Primary 52A21; Secondary 46B07, 52A40, 60D05.
DOI: https://doi.org/10.1090/proc12776
Published electronically: October 8, 2015
MathSciNet review: 3430852
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Abstract: A classical inequality of Rogers and Shephard states that if $K$ is a centered convex body of volume $1$ in ${\mathbb R}^n$, then \begin{equation*}1\leqslant g(K,k;F):=\big (\operatorname {vol}_k(P_F(K)) \operatorname {vol}_{n-k}(K\cap F^{\perp })\big )^{1/k} \leqslant {n\choose k}^{1/k}\leqslant \frac {cn}{k}\end{equation*} for every $F\in G_{n,k}$, where $c>0$ is an absolute constant. We show that if $K$ is origin symmetric and isotropic, then, for every $1\leqslant k\leqslant n-1$, a random $F\in G_{n,k}$ satisfies \begin{equation*}c_1L_K^{-1}\sqrt {n/k}\leqslant g(K,k;F)\leqslant c_2\sqrt {n/k}\ (\log n)^2 L_K\end{equation*} with probability greater than $1-e^{-k}$, where $L_K$ is the isotropic constant of $K$ and $c_1,c_2>0$ are absolute constants.


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Additional Information

Apostolos Giannopoulos
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
Email: apgiannop@math.uoa.gr

Eleftherios Markessinis
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
Email: lefteris128@yahoo.gr

Antonis Tsolomitis
Affiliation: Department of Mathematics, University of the Aegean, Karlovassi 832 00, Samos, Greece
MR Author ID: 605888
Email: antonis.tsolomitis@gmail.com

Keywords: Isotropic convex bodies, volume distribution, Rogers-Shephard inequality, isotropic constant
Received by editor(s): May 28, 2014
Received by editor(s) in revised form: November 19, 2014
Published electronically: October 8, 2015
Additional Notes: The authors would like to acknowledge support from the program “API$\Sigma$TEIA II – ATOCB – 3566” of the General Secretariat for Research and Technology of Greece.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society