Projections and Angle Sums of Belt Polytopes and Permutohedra

Abstract

AbstractLet $$P\subset \mathbb {R}^n$$ P ⊂ R n be a belt polytope, that is a polytope whose normal fan coincides with the fan of some hyperplane arrangement $${\mathcal {A}}$$ A . Also, let $$G:\mathbb {R}^n\rightarrow \mathbb {R}^d$$ G : R n → R d be a linear map of full rank whose kernel is in general position with respect to the faces of P. We derive a formula for the number of j-faces of the “projected” polytope GP in terms of the j-th level characteristic polynomial of $${\mathcal {A}}$$ A . In particular, we show that the face numbers of GP do not depend on the linear map G provided a general position assumption is satisfied. Furthermore, we derive formulas for the sum of the conic intrinsic volumes and Grassmann angles of the tangent cones of P at all of its j-faces. We apply these results to permutohedra of types A and B, which yields closed formulas for the face numbers of projected permutohedra and the generalized angle sums of permutohedra in terms of Stirling numbers of both kinds and their B-analogues.