  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.
