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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.21345 |
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| _version_ | 1866909597670834176 |
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| author | Jevtić, Filip D. Timotijević, Marinko Ž. Živaljević, Rade T. |
| author_facet | Jevtić, Filip D. Timotijević, Marinko Ž. Živaljević, Rade T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Indecomposability of the median hypersimplex and polytopality of the hemi-icosahedral Bier sphere Jevtić, Filip D. Timotijević, Marinko Ž. Živaljević, Rade T. Combinatorics Metric Geometry 52B12, 52B35, 52B70, 91A12 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. |
| title | Indecomposability of the median hypersimplex and polytopality of the hemi-icosahedral Bier sphere |
| topic | Combinatorics Metric Geometry 52B12, 52B35, 52B70, 91A12 |
| url | https://arxiv.org/abs/2504.21345 |