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 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 of \psi_{n} is equal to the cone given by the uniform cover inequalities.
We prove that this conjecture is nearly right: the closure of the convex hull of \psi_n is equal to the cone given by the uniform cover inequalities.
Joint work with Imre Leader and Zarko Randelovic.
