Many projectively unique polytopes

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.Comment: 44 pages, 18 figures; to appear in Invent. mat

An Efficient Local Approach to Convexity Testing of Piecewise-Linear Hypersurfaces

Abstract We show that a closed connected bounded piecewise-linear hypersurface in R n (n ≥ 3) is the boundary of a convex body if and only if every point in the interior of each (n − 3)-face has a neighborhood that lies on the boundary of some convex body; no assumptions about the hypersurface's topology are needed. We derive this criterion from our generalization of Van Heijenoort's (1952) theorem on locally convex hypersurfaces in R n to spherical spaces. We also give an easy-to-implement convexity testing algorithm, which is based on our criterion. For R 3 the number of arithmetic operations used by the algorithm is at most linear in the number of vertices, while in general it is at most linear in the number of incidences between the (n − 2)-faces and (n − 3)-faces. When the dimension n is not fixed and only ring arithmetic is allowed, the algorithm still remains polynomial. Our method works in more general situations than the convexity verification algorithms developed b

An Efficient Local Approach to Convexity Testing of Piecewise-Linear Hypersurfaces

Abstract We prove the following criterion: a compact connected piecewise-linear hypersurface (without boundary) in R n (n ≥ 3) is the boundary of a convex body if and only if every point in the relative interior of each (n−3)-face has a neighborhood that lies on the boundary of some convex body. This criterion is derived from our theorem that any connected complete locally-convex hypersurface in S n (n ≥ 3) is the boundary of a convex body in S n . We give an easy-to-implement convexity testing algorithm based on our criterion. This algorithm does not require any assumptions about the global topology of the input hypersurface. For R 3 the number of arithmetic operations used by our algorithm is at most linear in the number of vertices, while in general it is at most linear in the number of incidences between (n − 2)-faces and (n−3)-faces. The algorithm still remains polynomial even when the dimension n is a variable and the bit complexity model for (exact) arithmetic operations is used. The suggested method works in more general situations than the convexity verification algorithms developed b