In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the 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 $|T_A|$ 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 $A$. Bollobás 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 $\operatorname{conv}(\psi_n)$ is equal to the cone given by the uniform cover inequalities. We prove that this conjecture is “nearly” right: the closed convex hull $\overline{\operatorname{conv}}(\psi_n)$ is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that $\operatorname{conv}(\psi_n)$ is not closed for $n\ge 4$, thus disproving the conjecture.