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Convex Polytopes / Edition 2
     

Convex Polytopes / Edition 2

by Branko Grunbaum
 

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ISBN-10: 0387004246

ISBN-13: 9780387004242

Pub. Date: 05/12/2003

Publisher: Springer New York

"The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." —Peter

Overview

"The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." —Peter McMullen, University College London

Product Details

ISBN-13:
9780387004242
Publisher:
Springer New York
Publication date:
05/12/2003
Series:
Graduate Texts in Mathematics Series , #221
Edition description:
2nd ed. 2003
Pages:
471
Product dimensions:
6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

1 Notation and prerequisites.- 1.1 Algebra.- 1.2 Topology.- 1.3 Additional notes and comments.- 2 Convex sets.- 2.1 Definition and elementary properties.- 2.2 Support and separation.- 2.3 Convex hulls.- 2.4 Extreme and exposed points; faces and poonems.- 2.5 Unbounded convex sets.- 2.6 Polyhedral sets.- 2.7 Remarks.- 2.8 Additional notes and comments.- 3 Polytopes.- 3.1 Definition and fundamental properties.- 3.2 Combinatorial types of polytopes; complexes.- 3.3 Diagrams and Schlegel diagrams.- 3.4 Duality of polytopes.- 3.5 Remarks.- 3.6 Additional notes and comments.- 4 Examples.- 4.1 The d-simplex.- 4.2 Pyramids.- 4.3 Bipyramids.- 4.4 Prisms.- 4.5 Simplicial and simple polytopes.- 4.6 Cubical polytopes.- 4.7 Cyclic polytopes.- 4.8 Exercises.- 4.9 Additional notes and comments.- 5 Fundamental properties and constructions.- 5.1 Representations of polytopes as sections or projections.- 5.2 The inductive construction of polytopes.- 5.3 Lower semicontinuity of the functions fk(P).- 5.4 Gale-transforms and Gale-diagrams.- 5.5 Existence of combinatorial types.- 5.6 Additional notes and comments.- 6 Polytopes with few vertices.- 6.1 d-Polytopes with d + 2 vertices.- 6.2 d-Polytopes with d + 3 vertices.- 6.3 Gale diagrams of polytopes with few vertices.- 6.4 Centrally symmetric polytopes.- 6.5 Exercises.- 6.6 Remarks.- 6.7 Additional notes and comments.- 7 Neighborly polytopes.- 7.1 Definition and general properties.- 7.2 % MathType!MTEF!2!1!+-
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$$-Neighborly d-polytopes.- 7.3 Exercises.- 7.4 Remarks.- 7.5 Additional notes and comments.- 8 Euler’s relation.- 8.1 Euler’s theorem.- 8.2 Proof of Euler’s theorem.- 8.3 A generalization of Euler’s relation.- 8.4 The Euler characteristic of complexes.- 8.5 Exercises.- 8.6 Remarks.- 8.7 Additional notes and comments.- 9 Analogues of Euler’s relation.- 9.1 The incidence equation.- 9.2 The Dehn-Sommerville equations.- 9.3 Quasi-simplicial polytopes.- 9.4 Cubical polytopes.- 9.5 Solutions of the Dehn-Sommerville equations.- 9.6 The f-vectors of neighborly d-polytopes.- 9.7 Exercises.- 9.8 Remarks.- 9.9 Additional notes and comments.- 10 Extremal problems concerning numbers of faces.- 10.1 Upper bounds for fi, i— 1, in terms of fo.- 10.2 Lower bounds for fi, i— 1, in terms of fo.- 10.3 The sets f(P3) and f(PS3).- 10.4 The set fP4).- 10.5 Exercises.- 10.6 Additional notes and comments.- 11 Properties of boundary complexes.- 11.1 Skeletons of simplices contained in—(P).- 11.2 A proof of the van Kampen-Flores theorem.- 11.3 d-Connectedness of the graphs of d-polytopes.- 11.4 Degree of total separability.- 11.5 d-Diagrams.- 11.6 Additional notes and comments.- 12 k-Equivalence of polytopes.- 12.1 k-Equivalence and ambiguity.- 12.2 Dimensional ambiguity.- 12.3 Strong and weak ambiguity.- 12.4 Additional notes and comments.- 13 3-Polytopes.- 13.1 Steinitz’s theorem.- 13.2 Consequences and analogues of Steinitz’s theorem.- 13.3 Eberhard’s theorem.- 13.4 Additional results on 3-realizable sequences.- 13.5 3-Polytopes with circumspheres and circumcircles.- 13.6 Remarks.- 13.7 Additional notes and comments.- 14 Angle-sums relations; the Steiner point.- 14.1 Gram’s relation for angle-sums.- 14.2 Angle-sums relations for simplicial polytopes.- 14.3 The Steiner point of a polytope (by G. C. Shephard).- 14.4 Remarks.- 14.5 Additional notes and comments.- 15 Addition and decomposition of polytopes.- 15.1 Vector addition.- 15.2 Approximation of polytopes by vector sums.- 15.3 Blaschke addition.- 15.4 Remarks.- 15.5 Additional notes and comments.- 16 Diameters of polytopes (by Victor Klee).- 16.1 Extremal diameters of d-polytopes.- 16.2 The functions— and—b.- 16.3 Wv Paths.- 16.4 Additional notes and comments.- 17 Long paths and circuits on polytopes.- 17.1 Hamiltonian paths and circuits.- 17.2 Extremal path-lengths of polytopes.- 17.3 Heights of polytopes.- 17.4 Circuit codes.- 17.5 Additional notes and comments.- 18 Arrangements of hyperplanes.- 18.1 d-Arrangements.- 18.2 2-Arrangements.- 18.3 Generalizations.- 18.4 Additional notes and comments.- 19 Concluding remarks.- 19.1 Regular polytopes and related notions.- 19.2 k-Content of polytopes.- 19.3 Antipodality and related notions.- 19.4 Additional notes and comments.- Tables.- Addendum.- Errata for the 1967 edition.- Additional Bibliography.- Index of Terms.- Index of Symbols.

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