Inequalities on projected volumes

Abstract

In this paper we study the following geometric problem: given 2ⁿ − 1 real numbers x<inf>A</inf> indexed by the nonempty subsets A ⊂ {1,..., n}, is it possible to construct a body T ⊂ Rⁿ such that xA = |T<inf>A</inf>|, where |T<inf>A</inf>| 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 ψn the set of all vectors x for which there is a body T such that xA = log |T<inf>A</inf>| for all A. Bollobás and Thomason showed that ψn is contained in the polyhedral cone defined by the class of “uniform cover inequalities.” Tan and Zeng conjectured that the convex hull conv(ψn) is equal to the cone given by the uniform cover inequalities. We prove that this conjecture is “nearly” right: the closed convex hull conv(ψn) is equal to the cone given by the uniform cover inequalities. However, perhaps surprisingly, we also show that conv(ψn) is not closed for n ≥ 4, thus disproving the conjecture.