Reverse Loomis-Whitney inequalities via isotropicity

Given a centered convex body $K\subseteq {\mathbb{R}}^n$, we study the optimal value of the constant $\tilde{\mathrm{\Lambda }}(K)$ such that there exists an orthonormal basis $\{w_i{\}}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$\mathrm{\mid }K{\mathrm{\mid }}^{n-1}⩽\tilde{\mathrm{\Lambda }}(K)\prod _{i=1}^n\mathrm{\mid }K\cap w_i^{\perp }\mathrm{\mid }\mathrm{.}$$ We prove that $\tilde{\mathrm{\Lambda }}(K)⩽(C{L}_K{)}^n$ 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 $(c{L}_K{)}^n⩽\tilde{\mathrm{\Lambda }}(K)⩽(C{L}_K{)}^n$. We also prove more general reverse dual Loomis-Whitney inequalities as well as reverse restricted versions of Loomis-Whitney and dual Loomis-Whitney inequalities.