Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles
Metric Geometry
2019-10-10 v2
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
Cite
@article{arxiv.1812.00397,
title = {Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles},
author = {Filip D. Jevtić and Marinko Timotijević and Rade T. Živaljević},
journal= {arXiv preprint arXiv:1812.00397},
year = {2019}
}