Triangulations. Structures for algorithms and applications
This book masterfully presents the theory of triangulations of (the convex hull of) a point set alongside many appealing applications in algebra, computer science, combinatorics, and optimization. The authors ``firmly believe that understanding the fundamentals of geometry and combinatorics pays up for algorithms and applications'' of triangulations, and this interplay is beautifully developed. After introductory examples, setting up language and basic constructions, and a concrete chapter on computational geometry in dimension two, the book discusses Gelfand--Kapranov--Zelevinsky's central theorem on secondary polytopes (whose face lattices realize the posets of regular subdivisions of point configurations), followed by important families of configurations and triangulations (e.g., cyclic polytopes, product of simplices, 0/1-polytopes), a chapter on computation and algorithms, and a chapter on further topics (including fiber polytopes, mixed subdivisions, lattice polytopes, Gröbner bases and polytopal complexes). The writing is thorough and engaging, assisted by clear (and numerous) illustrations, and many exercises for the reader. Graduate students and researchers in any area in which triangulations of points set configurations play a role will find this book a comprehensive and most useful reference.
scientific article; zbMATH DE number 5768656