In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the non-empty subsets $Asubset {1,..,n}$, is it possible to construct a body $Tsubset mathbb{R}^n$ 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 $A$. Bollobas 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 $nge 4$, thus disproving the conjecture.
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