On the reverse Loomis-Whitney inequality

TL;DR

This paper investigates reverse Loomis-Whitney inequalities, focusing on estimating the LW-number for convex bodies, providing bounds, structural insights, and methods for computing the LW-constant of rational polytopes.

Contribution

It introduces new bounds and structural results for the reverse Loomis-Whitney inequality and addresses the computation of the LW-constant for rational polytopes.

Findings

Bounds on the LW-number $\lambda(n)$ for convex bodies.

Structural properties of reverse Loomis-Whitney inequalities.

Methods for computing the LW-constant of rational polytopes.

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 $\{u_1,\ldots,u_n\}$ such that $$ vol_n(K)^{n-1} \geq \gamma \prod_{i=1}^n vol_{n-1} (K|u_i^{\perp}) , $$ where $K|u_i^{\perp}$ 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.