The text under review is an introduction to some actual topics of the modern geometric combinatorics. Prepared for an advanced undergraduate course, it is aimed to present recent developments in the theory of convex polytopes, which are centered around secondary and state polytopes arising from point configurations.
The text consists of 14 chapters. Chapters 1–3 are devoted to some fundamental geometric and algebraic notions from the theory of polytopes. Chapters 4–6 deal with the representations of \(n\)-polytopes in terms of the well-known Schlegel diagrams and Gale diagrams. Point configurations in \(R^n\) and their triangulations are analyzed in details in Chapter 7. The next Chapter 8, the soul of the book, is devoted to the secondary polytopes of point configurations. As a concrete example of secondary polytopes, the permutahedron is described in Chapter 9. Fundamental relations between secondary polytopes and state polytopes of toric ideals of point configurations are established in Chapters 10–14. The theory of Gröbner bases, where the notion of state polytopes arises naturally, is developed in Chapters 10–12. Connections between Gröbner bases of toric ideals and regular triangulation of point configurations defining ideals are discussed in Chapter 13. In the final Chapter 14 the author demonstrates how to construct state polytopes of toric ideals naturally related to secondary polytopes. The theory of secondary and state polytopes is a recently developed direction of the modern geometry, which has numerous applications to combinatorics, algebraic geometry, discrete geometry, etc.
Moreover, the text itself is clearly written and self-contained, numerous illustrations and exercises are included. Thus, this “Lectures in geometric combinatorics” may be undoubtedly recommended for undergraduate and graduate students as a textbook on modern aspects of the theory of polytopes.