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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\u2019s 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 M\u00f6bius\u2019 torus with 7 vertices. Finally, we prove a partial result on Perles\u2019 problem whether all simplicial polytopes are quotients of neighborly polytopes.<\/jats:p>","DOI":"10.1137\/0401014","type":"journal-article","created":{"date-parts":[[2005,2,23]],"date-time":"2005-02-23T05:58:04Z","timestamp":1109138284000},"page":"121-133","source":"Crossref","is-referenced-by-count":15,"title":["Some Applications of Affine Gale Diagrams to Polytopes with Few Vertices"],"prefix":"10.1137","volume":"1","author":[{"given":"Bernd","family":"Sturmfels","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,8]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02937351"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5858-8_1"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(84)80033-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(78)90080-1"},{"key":"R5","unstructured":"J. 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