Many projectively unique polytopes

TL;DR

This paper constructs infinite families of polytopes with bounded realization space dimensions, solving longstanding problems about the relationship between polytope combinatorics and geometric realization uniqueness.

Contribution

It introduces new construction methods for projectively unique polytopes and answers key questions about realization space dimensions and projective uniqueness.

Findings

Constructed infinite families of 4-polytopes with small realization spaces.

Established the existence of 69-dimensional polytopes with unique projective realizations.

Developed novel techniques based on Cauchy problems, Alexandrov--van Heijenoort Theorem, and Lawrence's extension.

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.