Reverse Loomis-Whitney inequalities via isotropicity,Proceedings of the American Mathematical Society

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Reverse Loomis-Whitney inequalities via isotropicity
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-12-09 , DOI: 10.1090/proc/15265
David Alonso-Gutiérrez , Silouanos Brazitikos

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:

$\displaystyle \vert K\vert^{n-1}\leqslant \tilde {\Lambda }(K)\prod _{i=1}^n\vert K\cap w_i^\perp \vert.$

We prove that $\tilde {\Lambda }(K)\leqslant (CL_K)^n$ for some absolute

and that this estimate in terms of

, the isotropic constant of

, is asymptotically sharp in the sense that there exist another absolute constant

and a convex body

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.

中文翻译：

通过各向同性逆Loomis-Whitney不等式

摘要：给定一个中心凸体，我们研究常数的最佳值，以便存在一个正交的基础，并成立以下逆对偶Loomis-Whitney不等式： $K \ subseteq \ mathbb {R} ^ n$ $\ tilde {\ Lambda}（K）$ $\ {w_i \} _ {i = 1} ^ n$

$\ displaystyle \ vert K \ vert ^ {n-1} \ leqslant \ tilde {\ Lambda}（K）\ prod _ {i = 1} ^ n \ vert K \ cap w_i ^ \ perp \ vert。$

我们证明了一些绝对和这个估计来讲，的迷向常数，是在这个意义上渐近尖锐了存在另一个绝对常数和凸体这样。我们还证明了更一般的反向双重Loomis-Whitney不等式，以及反向限制性版本的Loomis-Whitney和双重Loomis-Whitney不等式。 $\ tilde {\ Lambda}（K）\ leqslant（CL_K）^ n$

$（cL_K）^ n \ leqslant \ tilde {\ Lambda}（K）\ leqslant（CL_K）^ n$

更新日期：2021-01-11

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