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 ${\mathbb{R}}^n$ there exists an orthonormal basis $\{u_1,\dots ,u_n\}$ such that $$vo{l}_n(K{)}^{n-1}\ge \gamma \prod _{i=1}^{n}vo{l}_{n-1}(K|u_i^{\perp }),$$ where $K|u_i^{\perp }$ denotes the orthogonal projection of $K$ onto the hyperplane $u_i^{\perp }$ 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.