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Classification of reflexive polyhedra in three dimensions. (English) Zbl 0934.52006
Given a lattice $$M$$, a lattice polytope is a polytope on the real extension $$M_{\mathbb R}$$ of $$M$$ whose vertices lie in $$M$$. A lattice polytope $$\Delta\subset M_{\mathbb R}$$ is called reflexive if its dual $$\Delta^* =\{ y\in N_{\mathbb R} : \langle y,x \rangle\geq -1\;\forall x\in M_{\mathbb R}\}$$ is a lattice polytope with respect to the lattice $$N$$ dual to $$M$$.
In the article under review, the authors present an algorithm for the classification of reflexive polytopes in arbitrary dimensions. They also present the results of an application of this algorithm to the case of three dimensional reflexive polytopes.
The study is motivated by the Calabi-Yau compactifications in string theory since the duality of reflexive polytopes corresponds to the mirror symmetry of the resulting Calabi-Yau manifolds. Previously the above mentioned algorithm was used by the authors to reobtain the known 16 two-dimensional reflexive polytopes [Commun. Math. Phys. 185, No. 2, 495-508 (1997; Zbl 0894.14026)].

##### MSC:
 52B15 Symmetry properties of polytopes 52B70 Polyhedral manifolds 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J30 $$3$$-folds
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