Indecomposability of the median hypersimplex and polytopality of the hemi-icosahedral Bier sphere

TL;DR

This work proves that the median hypersimplex $Δ_{2k,k}$ is Minkowski indecomposable by analyzing its essential deformation cone via a refinement by Bier spheres, establishing that Def_{ess}($Δ_{2k,k}$) is one-dimensional and hence yields a ray in the submodular cone. It also demonstrates that the Bier sphere of the hemi-icosahedron is polytopal in $\mathbb{R}^5$, providing explicit coordinates and facet data for the realizing polytope. The results connect the deformation theory of polytopes with polytopality questions for Bier spheres, advancing understanding of when certain combinatorial spheres admit convex realizations and clarifying the structure of deformed permutahedra via submodular and hypersimplex decompositions.

Abstract

We prove that the median hypersimplex $Δ_{2k,k}$ is Minkowski indecomposable, i.e. it cannot be expressed as a non-trivial Minkowski sum $Δ_{2k,k} = P+Q$, where $P\neq λΔ_{2k,k}\neq Q$. We obtain as a corollary that $Δ_{2k,k}$ represents a ray in the submodular cone (the deformation cone of the permutahedron). Building on the previously developed geometric methods and extensive computer search, we exhibit a twelve vertex, $4$-dimensional polytopal realization of the Bier sphere of the hemi-icosahedron, the vertex minimal triangulation of the real projective plane.

Figures (1)

• Figure 1: Hemi-icosahedron

Theorems & Definitions (19)

• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 3.1
• Remark 1
• Proposition 3.4
• Theorem 3.7
• Corollary 3.9
• Theorem 3.10
• proof