Reverse Loomis-Whitney inequalities via isotropicity

Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tildeΛ(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $|K|^{n-1}\leqslant \tildeΛ(K)\prod_{i=1}^n|K\cap w_i^\perp|.$ We prove that $\tildeΛ(K)\leqslant(CL_K)^n$ 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 $(cL_K)^n\leqslant\tildeΛ(K)\leqslant(CL_K)^n$. 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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