Projections and angle sums of belt polytopes and permutohedra

Abstract

Let $P\subset \mathbb R^n$ be a belt polytope, that is a polytope whose normal fan coincides with the fan of some hyperplane arrangement $\mathcal A$. Also, let $G:\mathbb R^n\to\mathbb R^d$ 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 $\mathcal A$. 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.