Geometry of Cuts and Metrics Cuts and metrics are well known objects that arise independently but with many deep and fascinating connections in diverse fields: in graph theory combinatorial optimization geometry of numbers distance geometry combinatorial matrix theory statistical physics VLSI design etc. A main feature of this book is its interdisciplinarity. The book contains a wealth of results from different mathematical disciplines which are presented here in a unified and comprehensive manner. Geometric representations and methods turn out to be the linking theme. This book will provide a unique and invaluable source for researchers and graduate students. From the Reviews: This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory geometry of numbers probability theory design and graph theory around two objects cuts and metrics. [… ] The book is very nicely written [… ] The book is also very well structured. With knowledge about the relevant terms one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. [… ] In my opinion the book is a beautiful piece of work. The longer one works with it the more beautiful it becomes. Robert Weismantel Optima 56 (1997) … In short this is a very interesting book which is nice to have. Alexander I. Barvinok MR 1460488 (98g:52001) … This is a large and fascinating book. As befits a book which contains material relevant to so many areas of mathematics (and related disciplines such as statistics physics computing science and economics) it is self contained and written in a readable style. Moreover the index bibliography and table of contents are all that they should be in such a work; it is easy to find as much or as little introductory material as needed. R.Dawson Zentralblatt MATH Database 0885.52001. Outline of the Book I.Measure Aspects: El Embeddability and Probability Preliminaries on Distances The Cut Cone and #x2113; Metrics The Correlation Cone and {0. 1} Covariances Conditions for Embeddability Operations Metrics from Lattices Semigroups and Normed Spaces Metric Transforms of Spaces Lipschitz Embeddings Dimensionality Questions for Embeddings Examples of the Use of the Metric Basic Definitions I1.Hypermetric Spaces: an Approach via Geometry of Numbers Preliminaries on Lattices Hypermetrics and Delaunay Polytopes Delaunay Polytopes: Rank and Hypermetric Faces Extreme Delaunay Polytopes Hypermetric Graphs I11.Isometric Embeddings of Graphs Preliminaries on Graphs Isometric Embeddings of Graphs into Hypercubes Isometric Embeddings of Graphs into Cartesian Products Graphs IV.Hypercube Embeddings and Designs Rigidity of the Equidistant Metric Hypercube Embeddings of the Equidistant Metric Recognition of Hypercube Embeddable Metrics Cut Lattices Quasi Distances and Hilbert Bases V.Facets of the Cut Cone and Polytope Operations on Valid Inequalities and Facets Triangle Inequalities Hypermetric Inequalities Clique Web Inequalities Other Valid Inequalities and Facets Geometric Properties.