Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

We consider families F (Δ) consisting of complex (n-1) -dimensional projective algebraic compactifications of Δ-regular affine hypersurfaces Zf defined by Laurent polynomials f with a fixed n-dimensional Newton polyhedron Δ in n-dimensional algebraic torus T = (C^*) ⁿ. If the family F (Δ) defined by a Newton polyhedron Δ consists of (n-1) -dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron Δ^* in the dual space defines another family F (Δ^*) of Calabi-Yau varieties, so that we obtain the remarkable duality between two different families of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of Mirror Symmetry discovered by physicists for Calabi-Yau 3-folds. Our method allows to construct many new examples of Calabi-Yau 3-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families F (Δ) and F (Δ^*).