Polytopes

The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject.
The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex.
With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes.
For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.

Proceedings of the NATO Advanced Study Institute, Scarborough, Ontario, Canada, August 20--September 3, 1993

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Abstract.- Recent results on Coxeter groups.- The evolution of Coxeter-Dynkin diagrams.- Polyhedra with hollow faces.- A hierarchical classification of Euclidean polytopes with regularity properties,.- Modern developments in regular polytopes.- Classification of locally toroidal regular polytopes.- Convex.- Face numbers and subdivisions of convex polytopes.- Approximation by convex polytopes.- Some aspects of the combinatorial theory of convex polytopes.- On volumes of non-Euclidean polytopes.- Manifolds in the skeletons of convex polytopes,tightness, and generalized Heawood inequalities.- Generalized stress and motions.- Polytopes and Brunn-Minkowski theory.- A survey of Eulerian posets.- Computational.- On recent progress in computational synthetic geometry.- The ridge graph of the metric polytope and some relatives.- On the complexity of some basic problems in Computational Convexity: II. Volume and mixed volumes.- The diameter of polytopes and related applications.- Problems.- Contributed problems.- Three problems about 4-polytopes.