A valuable resource for researchers in discrete and combinatorial geometry, this book offers comprehensive coverage of several modern developments on algebraic and combinatorial properties of polytopes. The introductory chapters provide a new approach to the basic properties of convex polyhedra and how they are connected; for instance, fibre operations are treated early on. Finite tilings and polyhedral convex functions play an important role, and lead to the new technique of tiling diagrams. Special classes of polytopes such as zonotopes also have corresponding diagrams. A central result is the complete characterization of the possible face-numbers of simple polytopes. Tools used for this are representations and the weight algebra of mixed volumes. An unexpected consequence of the proof is an algebraic treatment of Brunn–Minkowski theory as applied to polytopes. Valuations also provide a thread running through the book, and the abstract theory and related tensor algebras are treated in detail.