Convex Polytopes and Polyhedra

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.

Peter McMullen is Professor Emeritus of Mathematics at University College London. He was elected a foreign member of the Austrian Academy of Sciences in 2006, and a fellow of the American Mathematical Society in 2012. He is also a member of the London and European Mathematical Societies. He has co-edited several books, has co-authored 'Abstract Regular Polytopes' (Cambridge, 2002) and written 'Geometric Regular Polytopes' (Cambridge, 2020). His work has been discussed in the 'Encyclopaedia Britannica', and he was an invited speaker at the International Congress of Mathematicians in 1974.