On the reverse Loomis-Whitney inequality

Abstract: The present paper deals with the problem of computing (or at least estimating) the LW-number $\lambda(n)$, i.e., the supremum of all $\gamma$ such that for each convex body $K$ in $\mathbb{R}<sup>n$ there exists an orthonormal basis ${u_1,\ldots,u_n}$ such that $$vol_n(K)^{n-1} \geq \gamma \prod_{i=1}ⁿ vol_{n-1} (K|u_i^{\perp}) ,$$ where $K|u_i<sup>{\perp}$ denotes the orthogonal projection of $K$ onto the hyperplane $u_i<sup>{\perp}$ perpendicular to $u_i$. Any such inequality can be regarded as a reverse to the well-known classical Loomis--Whitney inequality. We present various results on such reverse Loomis--Whitney inequalities. In particular, we prove some structural results, give bounds on $\lambda(n)$ and deal with the problem of actually computing the LW-constant of a rational polytope.