On the Inequalities of Projected Volumes and the Constructible Region

We study the following geometry problem: given a $2^n-1$ dimensional vector $\pi=\{\pi_S\}_{S\subseteq [n], S\ne \emptyset}$, is there an object $T\subseteq\mathbb{R}^n$ such that $\log(\mathsf{vol}(T_S))= \pi_S$, for all $S\subseteq [n]$, where $T_S$ is the projection of $T$ to the subspace spanned by the axes in $S$? If $\pi$ does correspond to an object in $\mathbb{R}^n$, we say that $\pi$ is {\em constructible}. We use $\Psi_n$ to denote the constructible region, i.e., the set of all constructible vectors in $\mathbb{R}^{2^n-1}$. In 1995, Bollob\'{a}s and Thomason showed that $\Psi_n$ is contained in a polyhedral cone, defined a class of so called uniform cover inequalities. We propose a new set of natural inequalities, called nonuniform-cover inequalities, which generalize the BT inequalities. We show that any linear inequality that all points in $\Psi_n$ satisfy must be a nonuniform-cover inequality. Based on this result and an example by Bollob\'{a}s and Thomason, we show that constructible region $\Psi_n$ is not even convex, and thus cannot be fully characterized by linear inequalities. We further show that some subclasses of the nonuniform-cover inequalities are not correct by various combinatorial constructions, which refutes a previous conjecture about $\Psi_n$. Finally, we conclude with an interesting conjecture regarding the convex hull of $\Psi_n$.