Abstract
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a combinatorial type of 5-dimensional polytope that is not realizable as a subpolytope of any stacked polytope. This disproves a classical conjecture in polytope theory, first formulated by Shephard in the seventies.
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Supported by DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS UEFISCDI, project PN-II-ID-PCE-2011-3-0533.
Supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”.
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Adiprasito, K.A., Padrol, A. A universality theorem for projectively unique polytopes and a conjecture of Shephard. Isr. J. Math. 211, 239–255 (2016). https://doi.org/10.1007/s11856-015-1272-7
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DOI: https://doi.org/10.1007/s11856-015-1272-7