Projections and angle sums of belt polytopes and permutohedra

TL;DR

This paper develops formulas for the face counts and angle sums of projected belt polytopes and permutohedra, revealing invariance under certain linear maps and connecting geometric properties to combinatorial polynomials.

Contribution

It introduces a general formula linking projected belt polytopes' face numbers to the characteristic polynomial of hyperplane arrangements, and applies this to permutohedra for explicit combinatorial formulas.

Findings

Face numbers of projected belt polytopes depend only on arrangement, not the projection map.

Derived formulas for angle sums of tangent cones at faces.

Closed-form expressions for permutohedra face counts and angle sums using Stirling numbers.

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.