Authors: Jevtić, Filip 
Timotijević, Marinko
Živaljević, Rade 
Title: Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles
Journal: Discrete and Computational Geometry
Issue Date: 1-Jan-2019
Rank: M22
ISSN: 0179-5376
DOI: 10.1007/s00454-019-00151-5
Abstract: 
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”.
Keywords: Bier spheres | Gale transform | Kantorovich–Rubinstein polytopes | Polygonal linkages | Polyhedral combinatorics | Simplicial Steinitz problem
Publisher: Springer Link
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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