Some Applications of Affine Gale Diagrams to Polytopes with few Vertices

Affine Gale diagrams are of one dimension lower than the well-known Gale transforms, and thus k-polytopes with $k + 4$ vertices can be represented by planar point configurations. The underlying algebraic reduction is due to Bokowski [6], while similar geometric arguments were used before by Perles [12]. In this paper we consider affine Gale diagrams as a special case of oriented matroid duality, and we apply this technique to several convex geometrical problems. As the main result we establish a new negative Steinitz-type theorem in the spirit of [25]; the face lattices of simplicial k-polytopes with $k + 4$ vertices cannot be characterized locally. We answer two questions posed in [11] concerning Kleinschmidt’s 4-polytope Q with a facet of nonarbitrary shape [15], and we describe another such 4-polytope P with minimal number of facets. We characterize the affine Gale diagrams corresponding to a given simplicial complex, and we discuss as an example Mobius’ torus with 7 vertices. Finally, we prove a parti...