Abstract Factorization structures occur in toric differential and discrete geometry and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such or as an apex generalization of cyclic polytopes featuring a generalized Gale’s evenness condition. This article presents a comprehensive study of this new concept called factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains a construction of polytopes and cones compatible with a given factorization structure and exemplifies it for the product Segre–Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalized Gale’s evenness condition for compatible cones, polytopes, and their duals and explicitly describes faces of these. Factorization structures naturally provide generalized Vandermonde identities, which relate normals of any compatible polytope, and which are used to find examples of Delzant and rational Delzant polytopes compatible with the Veronese factorization structure. The article offers a myriad of factorization structure examples, which are later characterized to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.
Roland Púček (Fri,) studied this question.