On the inequalities of projected volumes and the constructible region

We study the following geometry problem: given a 2ⁿ − 1 dimensional vector π = {π_S}_S⊆[n],S6=∅, is there an object T ⊆ Rⁿ such that log(vol(T_S)) = π_S ∀S ⊆ [n], where T_S is the projection of T onto the subspace spanned by the axes in S and vol(T_S) is its |S|-dimensional volume? If π does correspond to an object in Rⁿ, we say that π is constructible. We use Ψn to denote the constructible region, i.e., the set of all constructible vectors in R²ⁿ⁻¹. In 1995, Bollobás and Thomason showed that Ψn is contained in a polyhedral cone and defined a class of so-called uniform-cover inequalities. We propose a new set of inequalities, called nonuniform-cover inequalities, which generalizes the uniform-cover inequalities. We show that any linear inequality that all points in Ψn satisfy must be a nonuniform-cover inequality. Based on this result and an example by Bollobás and Thomason, we show that the constructible region Ψn is nonconvex for n ≥ 4 and thus cannot be fully characterized by linear inequalities. We further show that some subclasses of the nonuniform-cover inequalities are not satisfied by all constructible vectors via various combinatorial constructions, which refutes a previous conjecture about Ψn. Finally, we conclude with an interesting conjecture regarding the convex hull of Ψn.