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arXiv · 1812.00397

Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles

Filip D. Jevtić·Marinko Timotijević·Rade T. Živaljević
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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".

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Filip D. Jevtić, Marinko Timotijević, Rade T. Živaljević. 2019-10-09. Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles. https://arxiv.org/abs/1812.00397

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