Inequalities on Projected Volumes

TL;DR

This paper investigates geometric inequalities related to the volumes of projections of bodies in n-dimensional space, proving a near-complete characterization of feasible volume log-values and disproving a longstanding conjecture about their convex hull.

Contribution

It proves that the closed convex hull of the set of feasible projection volume logs equals the cone defined by uniform cover inequalities, but the convex hull itself is not closed for four or more dimensions.

Findings

The closed convex hull of feasible volume logs matches the uniform cover inequalities cone.

The convex hull of feasible volume logs is not closed for n ≥ 4.

The conjecture that the convex hull equals the uniform cover inequalities cone is disproved.

Abstract

In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $A\subset \{1,..,n\}$, is it possible to construct a body $T\subset \mathbb{R}^n$ such that $x_A=|T_A|$ where $|T_A|$ is the $|A|$-dimensional volume of the projection of $T$ onto the subspace spanned by the axes in $A$? As it is more convenient to take logarithms we denote by $\psi_n$ the set of all vectors $x$ for which there is a body $T$ such that $x_A=\log |T_A|$ for all $A$. Bollob\'as and Thomason showed that $\psi_n$ is contained in the polyhedral cone defined by the class of `uniform cover inequalities'. Tan and Zeng conjectured that the convex hull $\DeclareMathOperator{\conv}{conv}$ $\conv(\psi_n)$ is equal to the cone given by the uniform cover inequalities. We prove that this conjecture is `nearly' right: the closed convex hull $\overline{\conv}(\psi_n)$ is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that $\conv (\psi_n)$ is not closed for $n\ge 4$, thus disproving the conjecture.