Lectures in Geometric Combinatorics
This book provides lectures in the geometry of convex polytopes in arbitrary dimension, with its geometric, combinatorial and computational aspects. This course is suitable for graduate students just starting out. The main goal of these lectures is to develop the theory of convex polytopes from a geometric viewpoint to lead up to recent developments centred around secondary and state polytopes arising from point configurations. The geometric viewpoint relies on linear optimization over polytopes. The book starts with the basics of the theory. Schlegel and Gale diagrams are introduced as tools to visualise polytopes in higher dimension. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the state polytope of the toric ideal defined by this configuration. There are numerous connections to discrete geometry, classical algebraic geometry and combinatorics. The connections rely on Groebner bases of toric ideals and other methods from commutative algebra. The lectures are self-contained and do not require any background beyond basic linear algebra.