Dual fans and mirror symmetry

We show that the mirror constructions of Greene–Plesser, Berglund–Hübsch, Batyrev, Batyrev–Borisov, Givental and Hori–Vafa can be expressed in terms of what we call dual fans. To do this, we associate to a pair of dual fans a pair of toric Landau–Ginzburg models, and we describe a process by which each of the mirror constructions listed also produces a pair of toric Landau–Ginzburg models. Replacing mirror pairs by toric Landau–Ginzburg models is reversible, and our main result is that the dual fan models and the mirror pairs models coincide.