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

We prove that the median hypersimplex ${\mathrm{\Delta }}_{2k,k}$ is Minkowski indecomposable, i.e. it cannot be expressed as a non-trivial Minkowski sum ${\mathrm{\Delta }}_{2k,k}=P+Q$, where $P\mathrm{\ne }\lambda {\mathrm{\Delta }}_{2k,k}\mathrm{\ne }Q$. We obtain as a corollary that ${\mathrm{\Delta }}_{2k,k}$ 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.