Many polytopes with low-dimensional realization space
Karim Adiprasito and Günter M. Ziegler
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: To bound the dimension of the realization space of a polytope in terms of its f-vector. Moreover, 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.