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Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles


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”.

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This research was supported by the Grants 174020 and 174034 of the Ministry of Education, Science and Technological Development of the Republic of Serbia. It is our pleasure to acknowledge valuable remarks and useful suggestions by Bernd Sturmfels, Günter M. Ziegler, Siniša Vrećica, Duško Jojić, Vladimir Grujić, members of Belgrade CGTA-seminar, and the anonymous referees.

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Correspondence to Rade T. Živaljević.

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Jevtić, F.D., Timotijević, M. & Živaljević, R.T. Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles. Discrete Comput Geom 65, 1275–1286 (2021).

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  • Kantorovich–Rubinstein polytopes
  • Gale transform
  • Bier spheres
  • Polyhedral combinatorics
  • Simplicial Steinitz problem
  • Polygonal linkages

Mathematics Subject Classification

  • 52B12
  • 52B35
  • 52B70