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On the classification of reflexive polyhedra. (English) Zbl 0894.14026
The Calabi-Yau condition for hypersurfaces of toric varieties is known to be equivalent to the reflexivity of a certain underlying polyhedron defined on an integer lattice. Moreover, the duality of reflexive polyhedra corresponds in this association to the mirror symmetry of the resulting Calabi-Yau manifolds. The authors investigate the geometrical structures of minimal polytopes containing the original polytope, i.e. circumscribed polytopes with a minimal number of facets and inscribed polyhedra with a minimal number of vertices. These objects, which put constraints on reflexive pairs of polyhedra, can be described in terms of certain non-negative integral matrices. A classification algorithm is deduced. This algorithm is proven to be finite and is used to reobtain the known 16 2-dimensional polyhedra. It is also expected to be efficient enough for a classification of polyhedra in dimensions less or equal to 4, this last dimension being relevant for the Calabi-Yau compactifications in string theory.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B15 Symmetry properties of polytopes
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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