A Conjectural Non-commutative Generalization of a Volume Formula of Mcmullen-schneider

Let V be a real vector space of dimension d and V ∗ its dual space. By a cone in V ∗ we will always mean a closed polyhedral cone σ with apex 0 such that σ ∩ −σ = {0}. Let Σ be a fan in V ∗, i.e., a collection of cones such that (1) if σ ∈ Σ then any face of σ belongs to Σ, (2) if σ1, σ2 ∈ Σ then σ1 ∩ σ2 is a face in both. We will assume that Σ is complete, that is ∪Σ = V ∗. The elements of Σ are called faces. We denote by Σ(i) the set of i-dimensional faces of Σ. In particular, Σ(d), Σ(d − 1) and Σ(1) are the sets of chambers, walls and rays of Σ respectively. Two chambers are adjacent if they intersect in a wall. Any wall is contained in exactly two chambers (which are adjacent). We will write σ τ ←→ σ̃ if σ and σ̃ are adjacent with common wall τ = σ ∩ σ̃. If we want to distinguish σ we will write σ τ −→ σ̃ and speak of a directed wall ω emerging from σ. We denote by ω̃ the opposite directed wall σ̃ τ −→ σ. Henceforth, we will assume that Σ is simplicial, that is, each cone in Σ is simplicial. Equivalently, any chamber σ has precisely d directed walls emerging from it. For any cone σ we denote by V(σ) its linear span and by σ⊥ its annihilator in V . A d-tuple (τ1, . . . , τd) of walls is called transversal if ∑d i=1 τ ⊥ i = V , i.e. if ∩i=1V(τi) = 0. A basic example of a simplicial fan is the normal fan ΣP of a simple convex polytope P in V , whose affine hull is V . It is given by ΣP = {τ(F ) : F ∈ F(P )} where F(P ) denotes the lattice of faces of P and