Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles. (English) Zbl 1462.52019
Summary: The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the ‘Simplicial Steinitz problem’. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of “short sets”.
MSC:
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
| 52B35 | Gale and other diagrams |
| 52B70 | Polyhedral manifolds |
| 57Q15 | Triangulating manifolds |
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
Keywords:
Kantorovich-Rubinstein polytopes; gale transform; Bier spheres; polyhedral combinatorics; simplicial Steinitz problem; polygonal linkagesReferences:
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