We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to (Formula presented.). This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its (Formula presented.)-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in (Formula presented.), a new Alexandrov–van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence’s extension technique for point configurations.
Bibliographical noteFunding Information:
This work was supported by the 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. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels and the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”.
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