Convex Polytopes

Convex Polytopes stands as a foundational graduate-level mathematics text examining the properties of convex polytopes and their higher-dimensional applications. First published in 1967 and rereleased in 2003, this text by Branko Grünbaum includes contributions from distinguished mathematicians Victor Klee, Micha Perles, and G.C. Shephard. The book progresses through 19 chapters, beginning with essential background in linear algebra and topology before advancing to complex geometric concepts. Core topics include polyhedra definitions, Schlegel diagrams, Gale diagrams, and cyclic polytopes, with each concept building upon previous material. The text earned the 2005 Leroy P. Steele Prize for mathematical exposition from the American Mathematical Society. The Mathematical Association of America endorsed its inclusion in undergraduate mathematics libraries, establishing its position as a key resource in geometric theory. Beyond its technical content, the book represents a significant advancement in the field of geometric mathematics, bridging fundamental concepts with advanced theoretical frameworks. Its systematic approach to polytope theory creates connections between various mathematical disciplines.

Readers note this is a dense, technical reference work that requires significant mathematical background. Multiple reviewers call it the definitive text on polytope theory but warn it's not suitable as a first introduction to the subject. Positives: - Comprehensive coverage of polytope properties and theorems - Clear notation and precise definitions - High quality illustrations and diagrams - Thorough bibliography and historical notes Negatives: - Advanced prerequisites in linear algebra and geometry - Limited worked examples - Some readers found the layout and typesetting dated Available ratings: Goodreads: 4.8/5 (5 ratings) Amazon: Not enough reviews for rating Google Books: No ratings Direct quote from mathematician David Eppstein's review: "This remains the standard reference on convex polytopes, despite its age. The rigorous treatment and careful attention to detail make it invaluable for research, though beginners should start elsewhere." [Note: Limited review data available since this is a specialized academic text]

Lectures on Polytopes by Günter M. Ziegler This text provides complementary perspectives on polytope theory with focus on face numbers and Dehn-Sommerville relations.

Theory of Linear and Integer Programming by Alexander Schrijver The book connects polytope theory to optimization through detailed examination of polyhedra in linear programming.

Handbook of Discrete and Computational Geometry by Joseph O'Rourke This reference work includes extensive sections on polytopes and their applications in computational geometry.

Polytopes, Rings and K-Theory by Winfried Bruns, Joseph Gubeladze The text explores connections between polytope theory and commutative algebra through toric geometry.

Regular Polytopes by H.S.M. Coxeter This classic work focuses on regular polytopes in multiple dimensions with emphasis on symmetry groups and classifications.

★ The original 1967 edition of "Convex Polytopes" is considered a landmark text that helped establish modern polytope theory as a distinct mathematical discipline. ★ Some diagrams in the book were hand-drawn by Grünbaum's wife, Zdenka, who was credited for her meticulous illustrations that helped readers visualize complex geometric concepts. ★ Grünbaum introduced several innovative visualization techniques, including the systematic use of Schlegel diagrams, which project higher-dimensional polytopes onto lower dimensions while preserving their structure. ★ The book's influence extends beyond mathematics into fields like theoretical physics and computer graphics, where its principles are applied in understanding crystal structures and 3D modeling. ★ The 2003 second edition added over 200 pages of new material, incorporating major developments in polytope theory that occurred during the 35-year gap between editions.