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^*).