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Oriented matroids. (English) Zbl 0773.52001
Encyclopedia of Mathematics and Its Applications. 46. Cambridge: Cambridge University Press. 516 p. (1993).
The book seems to be the first comprehensive exposition of the field. Oriented matroids have applications in many different areas which include among others disrete and computational geometry, convexity, topology, operations research, and theoretical chemistry. The first two chapters give a short introduction to the field of interest. Then the following topics are discussed: Axiomatics, face lattices and topology, topological models for oriented matroids, arrangements of pseudolines, constructions (including single element extensions, lexicographic extensions, local perturbations and mutations, intersection properties, direct sum and union, strong maps and weak maps, inseparability graphs, and orientability), realizability (including among others the realization space of an oriented matroid, the isotopy problem, Mnëv’s universality theorem and a discussion about robust computational geometry), convex polytopes (matroid polytopes, the Lawrence construction, cyclic and neighborly matroid polytopes, the Steinitz problem, and subdivisions and triangulations) and linear programming.
The book contains several interesting new results. A list of problems and exercises is included after each of the ten chapters. Many references are given at the end of the book, which is a good introduction for graduate students and a thorough reference work for specialists also.
Reviewer: H.-D.Hecker (Jena)

52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05B35 Combinatorial aspects of matroids and geometric lattices
90C05 Linear programming
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)