1995/12/31 by Maximilian Kreuzer, M. Kreuzer, Harald Skarke +1
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Combinatorics #Commutative Algebra and Its Applications #Dual polyhedron #Geometry #Inscribed figure #Mathematics #Polyhedron #Polytope #Pure mathematics #Reflexivity #String (physics) #alg-geom #hep-th #math.AG
paper · pdf · doi:10.1007/s002200050100
published as Commun.Math.Phys. 185 (1997) 495-508 · 13 pages, LaTeX (updated to agree with shortened published version)
arxiv created 1997/02/04 · openalex publication_date 1997/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the corner stone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi-Yau compactifications in string theory.