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

Abstract

We prove that the median hypersimplex ∆2k,k is Minkowski indecomposable, i.e. it cannot be expressed as a non-trivial Minkowski sum ∆2k,k = P + Q, where P , λ∆2k,k , Q. Since ∆2k,k is a deformed permutahedron, we obtain as a corollary that ∆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.