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arXiv · 2504.21345

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

Filip D. Jevtić·Marinko Ž. Timotijević·Rade T. Živaljević
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Abstract

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

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Filip D. Jevtić, Marinko Ž. Timotijević, Rade T. Živaljević. 2025-04-30. Indecomposability of the median hypersimplex and polytopality of the hemi-icosahedral Bier sphere. https://arxiv.org/abs/2504.21345

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