# Many projectively unique polytopes

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## Abstract

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to \(96\). 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 \(f\)-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 \(S^d\), a new Alexandrov–van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence’s extension technique for point configurations.

## Notes

### Acknowledgments

We are grateful to Alexander Bobenko for background information on Q-nets, to Igor Pak for valuable discussions, to Francisco Santos for particularly helpful suggestions, and to Miriam Schlöter for some of the figures in this paper. The first author thanks the Hebrew University Jerusalem, whose hospitality he enjoyed on several occasions while working on this paper.

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