2009/08/31 by Sven Herrmann · 11 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Birkhoff polytope #Combinatorics #Commutative Algebra and Its Applications #Geometry #Mathematics #Polynomial and algebraic computation #Polytope #Regular polygon #math.CO #math.MG #msc:52B11 #msc:52B30 #msc:52B35 #msc:52B40
paper · pdf · doi:10.1016/j.jcta.2010.08.003
published in Journal of Combinatorial Theory Series A 118(2), 425-447 (Elsevier BV) · 28 pages, 18 figures
arxiv created 2010/07/26 · openalex publication_date 2010/09/01 · arxiv updated 2014/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.