Factorization structures, cones, and polytopes

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 K\"ahler geometries, as special coordinates on such, or as an apex generalisation of cyclic polytopes featuring a generalised Gale's evenness condition. This article presents a comprehensive study of factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains the construction of polytopes and cones compatible with a given factorization structure, and exemplifies it for product Segre-Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalised Gale's evenness condition for compatible cones, polytopes and their duals, and explicitly describes faces of these. Factorization structures naturally provide generalised Vandermonde identities, which relate normals of any compatible polytope, and which are used for Veronese factorization structure to find examples of Delzant and rational Delzant compatible polytopes. The article offers a myriad of factorization structure examples, which are later characterised 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.