Reverse Loomis-Whitney inequalities via isotropicity

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: $$|K{|}^{n-1}⩽\tilde{\Lambda }(K)\prod _{i=1}^n|K\cap w_i^{\perp }|.$$ We prove that $\tilde{\Lambda }(K)⩽(C{L}_K{)}^n$ 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 exists another absolute constant $c>1$ and a convex body $K$ such that $(c{L}_K{)}^n⩽\tilde{\Lambda }(K)⩽(C{L}_K{)}^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.