The book "Geometric Algorithms and Combinatorial Optimization" is a second corrected edition, edited by Martin Grötschel, László Lovász, and Alexander Schrijver. It presents a comprehensive overview of geometric algorithms and combinatorial optimization, covering both theoretical and practical aspects. The book discusses the ellipsoid method, a geometric algorithm for solving linear programming problems in polynomial time, and the geometry of numbers, which has deep applications in number theory and integer programming. It also explores the use of basis reduction in lattice algorithms and its applications in combinatorial optimization. The book is structured into chapters that cover mathematical preliminaries, complexity theory, oracles, numerical computation, and the algorithmic aspects of convex sets. It includes detailed discussions on the ellipsoid method, algorithms for convex bodies, diophantine approximation, and rational polyhedra. The text also addresses various combinatorial optimization problems, such as flows, cuts, matchings, and submodular functions, and their solutions using geometric and combinatorial methods. The authors emphasize the theoretical framework for polynomial time solvability of combinatorial optimization problems, highlighting the equivalence of problems that are dual in various senses. They also discuss the limitations of geometric methods in comparison to specialized algorithms, focusing on the polynomial time solvability rather than the best possible running times. The book includes a detailed table of contents, with chapters covering topics such as mathematical preliminaries, complexity, oracles, convex sets, the ellipsoid method, algorithms for convex bodies, diophantine approximation, rational polyhedra, combinatorial optimization examples, and submodular functions. It is intended for researchers and students in combinatorial optimization, computational convexity, and related fields. The book is supported by a comprehensive list of references, notation index, author index, and subject index.The book "Geometric Algorithms and Combinatorial Optimization" is a second corrected edition, edited by Martin Grötschel, László Lovász, and Alexander Schrijver. It presents a comprehensive overview of geometric algorithms and combinatorial optimization, covering both theoretical and practical aspects. The book discusses the ellipsoid method, a geometric algorithm for solving linear programming problems in polynomial time, and the geometry of numbers, which has deep applications in number theory and integer programming. It also explores the use of basis reduction in lattice algorithms and its applications in combinatorial optimization. The book is structured into chapters that cover mathematical preliminaries, complexity theory, oracles, numerical computation, and the algorithmic aspects of convex sets. It includes detailed discussions on the ellipsoid method, algorithms for convex bodies, diophantine approximation, and rational polyhedra. The text also addresses various combinatorial optimization problems, such as flows, cuts, matchings, and submodular functions, and their solutions using geometric and combinatorial methods. The authors emphasize the theoretical framework for polynomial time solvability of combinatorial optimization problems, highlighting the equivalence of problems that are dual in various senses. They also discuss the limitations of geometric methods in comparison to specialized algorithms, focusing on the polynomial time solvability rather than the best possible running times. The book includes a detailed table of contents, with chapters covering topics such as mathematical preliminaries, complexity, oracles, convex sets, the ellipsoid method, algorithms for convex bodies, diophantine approximation, rational polyhedra, combinatorial optimization examples, and submodular functions. It is intended for researchers and students in combinatorial optimization, computational convexity, and related fields. The book is supported by a comprehensive list of references, notation index, author index, and subject index.