• norm
• unit ball
• inequality
• fibered faces
• norma
• palla unitaria
• disuguaglianza
• facce fibrate
• Thurston
• Alexander
• McMullen

The objective of this thesis is to prove McMullen's inequality for M a compact, connected, orientable 3-manifold, whose boundary, if any, is a union of tori.
The inequality bounds the Alexander norm of any class in H^1(M; Z) with the Thurston norm, up to a small correction term if the first Betti number of M is 1.
After an overview of basic facts about the Thurston norm and its unit ball, the Alexander module, ideal and polynomial are defined, to enable the study of the Alexander norm of a 3-manifold. Then, the main inequality is proven, following McMullen's original article.
Finally, some examples are described, with particular attention to the equality cases, as shown in Dunfield's famous example of strict containment between faces of the two unit balls.