Some applications of affine Gale diagrams to polytopes with few vertices

摘要: Affine Gale diagrams are of one dimension lower than the well-known 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 Perles [12]. In this paper we consider affine as a special case oriented matroid duality, apply technique several convex geometrical problems. As main result establish new negative Steinitz-type theorem in spirit [25]; face lattices simplicial cannot characterized locally. We answer two questions posed [11] concerning Kleinschmidt’s 4-polytope Q facet nonarbitrary shape [15], describe another such P minimal number facets. characterize corresponding given complex, discuss an example Mobius’ torus 7 vertices. Finally, prove parti...