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: \[ |K|^{n-1}\leqslant \tilde{\Lambda}(K)\prod_{i=1}^n|K\cap w_i^\perp|. \] 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 exists 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.