Many projectively unique polytopes

Karim A. Adiprasito, Günter M. Ziegler*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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.

Original languageAmerican English
Pages (from-to)581-652
Number of pages72
JournalInventiones Mathematicae
Issue number3
StatePublished - Mar 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.


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