ON THE CLASSIFICATION OF REFLEXIVE POLYHEDRA

Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi-Yau hypersurfaces in toric varieties. We investigate the geomet rical structures of circumscribed polytopes with a minimal number of f acets and of inscribed polytopes with a minimal number of vertices. Th ese objects, which constrain reflexive pairs of polyhedra from the int erior and the exterior, can be described in terms of certain non-negat ive integral matrices. A major tool in the classification of these mat rices is the existence of a pair of weight systems, indicating a relat ion to weighted projective spaces. This is the cornerstone for an algo rithm for the construction of all dual pairs of reflexive polyhedra th at we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi-Yau compa ctifications in string theory.