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Reverse Loomis-Whitney inequalities via isotropicity

Authors: David Alonso-Gutiérrez and Silouanos Brazitikos
Journal: Proc. Amer. Math. Soc. 149 (2021), 817-828
MSC (2010): Primary 52A23; Secondary 60D05
Published electronically: December 9, 2020
MathSciNet review: 4198086
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Abstract: Given a centered convex body $K\subseteq \mathbb {R}^n$, we study the optimal value of the constant $\tilde {\Lambda }(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: \begin{equation*} |K|^{n-1}\leqslant \tilde {\Lambda }(K)\prod _{i=1}^n|K\cap w_i^\perp |. \end{equation*} We prove that $\tilde {\Lambda }(K)\leqslant (CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exist another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant \tilde {\Lambda }(K)\leqslant (CL_K)^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.

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David Alonso-Gutiérrez
Affiliation: Área de Análisis Matemático, Departamento de Matemáticas, Facultad de Ciencias, IUMA, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
ORCID: 0000-0003-1256-3671

Silouanos Brazitikos
Affiliation: School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, JCMB, Peter Guthrie Tait Road King’s Buildings, Mayfield Road, Edinburgh, EH9 3FD, Scotland
MR Author ID: 1059952

Received by editor(s): January 29, 2020
Received by editor(s) in revised form: June 9, 2020
Published electronically: December 9, 2020
Additional Notes: The first author was partially supported by MINECO Project MTM2016-77710-P, DGA E48_27R, and MICINN PID2019-105979GB-I00.
The second author was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (Project Number: 1849).
Communicated by: Deane Yang
Article copyright: © Copyright 2020 American Mathematical Society