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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.
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.
Regular polytopes and their symmetry have a long history stretching
back two and a half millennia, to the classical regular polygons
and polyhedra. Much of modern research focuses on abstract regular
polytopes, but significant recent developments have been made on
the geometric side, including the exploration of new topics such as
realizations and rigidity, which offer a different way of
understanding the geometric and combinatorial symmetry of
polytopes. This is the first comprehensive account of the modern
geometric theory, and includes a wide range of applications, along
with new techniques. While the author explores the subject in
depth, his elementary approach to traditional areas such as finite
reflexion groups makes this book suitable for beginning graduate
students as well as more experienced researchers.
Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area since Coxeter's "Regular Polytopes" (1948) and "Regular Complex Polytopes" (1974).
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