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Lectures on Polytopes

G. M. Ziegler
Springer Verlag
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 152
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[Reviewed by
Miklós Bóna
, on

From the publication of the first printing, in 1994, this book became one of the most widely used textbooks in Discrete Geometry. This reviewer sees at least two reasons for that: the beautiful mathematics presented here, and the fact that the book can be used at a wide variety of levels, for several different courses.

Chapters 0–3 contain very general introductory (but interesting and elegant) material that every discrete mathematician should read. This part ends with two strong results. One is Gil Kalai's simple method to tell a simple polytope from its graph, and the other is Balinski's theorem stating that the graph of a d-dimensional polytope is d-connected.

After this, there are several directions an instructor can take. If the class consists of exceptionally motivated students, covering most of the book is not unrealistic. Otherwise, one can go in the direction of Polytopes and Polyhedral complexes, covering chapters 4, 5, and 9. This will start with the famous theorem of Steinitz claiming that a graph is the graph of a simple polyhedron if and only if it is simple, planar and 3-connected, and end with constructing the permuto-associohedron.

Another possibility is to turn the course into a special topics course on duality, zonotopes, and oriented matroids, which would involve covering Chapters 6 and 7. Chapter 8 is on Shellability and the Upper Bound Theorem. It discusses several classic results, such as the fact that polytopes are shellable, Dean-Somerville equations, and the Kruskal-Katona theorem from extremal set theory.

It is not only students who can benefits from the book. Researchers will find its updated notes and references very helpful. This is a good place to mention what "Updated Seventh Printing" means. Throughout these seven printings, the author has left the book's structure, and even the page numbering (!) intact. On the other hand, he has made a series of small changes, corrections, and continuously updated the notes and the references.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida

 Preface; Preface to the Second Printing; Introduction and Examples; 1. Polytopes, Polyhedra, and Cones; 2. Faces of Polytopes; 3. Graphs for Polytopes; 4. Steinitz' Theorem for 3-Polytopes; 5. Schlegel Diagrams for 4-Polytopes; 6. Duality, Gale Diagrams, and Applications; 7. Fans, Arrangements, Zonotopes, and Tilings; 8. Shellability and the Upper Bound Theorem; 9. Fiber Polytopes, and Beyond; References; Index.