Projections and angle sums of belt polytopes and permutohedra

2020/09/09 by Thomas Godland, Godland, Thomas, Zakhar Kabluchko +1
Engineering · Mathematics · #51F15 #52A22 #52A55 (Secondary) #52B05 #52B11 #60D05 (Primary) 11B73 #Advanced Combinatorial Mathematics #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2009.04186

openalex publication_date 2020/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P⊂ \mathbb Rⁿ 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 Rⁿ→\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.

Related

• Projection volumes of hyperplane arrangements
• Angle sums on polytopes and polytopal complexes
• Faces of Generalized Permutohedra
• Vector spaces spanned by the angle sums of polytopes
• Angle sums of random polytopes