Inequalities on Projected Volumes

In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $A\subset \{1,\mathrm{.}\mathrm{.},n\}$, is it possible to construct a body $T\subset {\mathbb{R}}^n$ such that $x_A=\mathrm{\mid }{T}_A\mathrm{\mid }$ where $\mathrm{\mid }{T}_A\mathrm{\mid }$ is the $\mathrm{\mid }A\mathrm{\mid }$-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 $x_A=\log \mathrm{\mid }{T}_A\mathrm{\mid }$ 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 $\overline{\text{conv}}({\psi }_n)$ is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that $\text{conv}({\psi }_n)$ is not closed for $n\ge 4$, thus disproving the conjecture.