Inequalities on Projected Volumes

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