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Grünbaum B. — Convex Polytopes
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Название: Convex Polytopes
Автор: Grünbaum B.
Аннотация: The appearance of Gruenbaum's book Convex Polytopes in 1967 was a moment of grace to geometers and combinatorialists. The special spirit of the book is very much alive even in those chapters where the book's immense influence made them quickly obsolete. Some other chapters promise beautiful unexplored land for future research. The appearance of the new edition is going to be another moment of grace. Kaibel, Klee and Ziegler were able to update the convex polytope saga in a clear, accurate, lively, and inspired way. —Gil Kalai, The Hebrew University of Jerusalem The original book of Gruenbaum has provided the central reference for work in this active area of mathematics for the past 35 years...I first consulted this book as a graduate student in 1967; yet, even today, I am surprised again and again by what I find there. It is an amazingly complete reference for work on this subject up to that time and continues to be a major influence on research to this day. —Louis J. Billera, Cornell University The original edition of Convex Polytopes 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 LondonThe combinatorial study of convex polytopes is today an extremely active and healthy area of mathematical research, and the number and depth of its relationships to other parts of mathematics have grown astonishingly since Convex Polytopes was first published in 1966. The new edition contains the full text of the original and the addition of notes at the end of each chapter. The notes are intended to bridge the thirty five years of intensive research on polytopes that were to a large extent initiated, guided, motivated and fuelled by the first edition of Convex Polytopes. The new material provides a direct guide to more than 400 papers and books that have appeared since 1967. Branko Grünbaum is Professor of Mathematics at the University of Washington.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Издание: Second Edition
Год издания: 2003
Количество страниц: 466
Добавлена в каталог: 30.06.2008
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Предметный указатель
Polytope, shellability 142a
Polytope, simple see “Simple polytope”
Polytope, simplicial see “Simplicial polytope”
Polytope, spherical 7a
Polytope, stacked 198b
Polytope, summand 316
Polytope, type (2,2) 69a 82 96b
Polytope, type (2,d-2) 171b 428b
Polytope, type (3,d-3) 171b
Polytope, type (4,4) 69a
Polytope, type (k,h) 58 65 69a 169
Polytope, type (r+2,s+t-1) 171b
Polytope, unavoidable small faces 224b
Polytope, uniform 171b 413 423a
Polytopes, enumeration of see “Enumeration”
Polytopes, number of see “Number”
Polytopes, related 50
Poonem 20
Poonem, not being a face 30a
Poonem, polyhedral sets 26
Positively homogeneous function 13
Primitive fixing system 423
Primitive illuminating set 422
Primitive polytope 423
Prism, d-prism 56
Prism, k-fold d-prism 56
Prismoid 57
Prismoid, generalized 65
Prismoid, indecomposability 323
Prismoid, proof of Euler’s theorem 131
Projection constant 73
Projection of a polytope 71
Projection, radial 23
Projective equivalence 5
Projective equivalence, Gale-transforms 89
Projective geometry 7b
Projective space 4
Projective space, arrangements 390
Projective space, configurations 93
Projective space, Euler characteristic 142
Projective transformation 4 7b
Projective transformation of convex sets 21 35
Projective transformation of d+3 points in 126
Projective transformation of polytopes 67
Projective transformation, admissible 7a
Projective transformation, Gale-diagrams 87
Projective transformation, permissible 4
Projective transformation, piecewise 41
Projective transformation, polytopes inscribed into spheres 285
Projectively convex set 29
Projectively realizable sequence 405
Projectively regular polytope 412
Projectively unique 68
Projectively unique, d-polytopes with d+3 vertices 120
Projectively unique, non-rational 8-polytope 96
Projectively unique, polytopes with not, facets 208
Pseudo-hyperplane arrangement 410a
Pseudo-hyperplane arrangement, number of simplicial regions 410b
Pseudo-line 408
Pseudo-line arrangement 410a
Pseudo-line arrangement, number of triangles 410b
Pulling 82 96b
Pushing 82 96b
Pyramid 54
Pyramid, r-fold 54
Pyramidal polytope 68 88
Pyramidoid 63
Pyramidoid, indecomposability 323
Pyramids, f-vectors of r-fold 140
Pyramids, indecomposability of r-fold 323
Pyramids, r-fold, and d-polytopes with d+2 vertices 100
Pyramids, r-fold, and pyramidoids 64
Quasi-polyhedral 36
Quasi-simplicial polytope 59
Quasi-simplicial polytope, Euler hyperplane 137
Quasi-simplicial polytope, f-vectors 153
Radial projection 23 200
Radius of a polytope 341
Radon’s theorem 16
Radon’s theorem, k-neighborly polytopes 123
Ramsey theory 7b 30b
Ramsey’s theorem 22 126
Random walk 340b
Random-edge pivot rule 389b
Random-facet pivot rule 389b
Rational polytope 52a 92
Rational polytope, decidability of face lattices 96b
Rational polytope, perturbing a polytope with d+3 vertices 119
Rational polytope, Steinitz’ theorem 244
Rational space, polytopes 92
Rational space, sections 76
Ray shooting 52a
readability see “d-realizability”
Readability of 2-complexes 253
Readability, -readability, of a complex see “ -realizable”
Readability, -readability, of a complex see “ -realizable”
Readability, 3-readability, of a sequence see “3-realizable sequence”
Readability, projective, of a sequence see “Projectively realizable sequence”
Reconstruction of duals of capped cubical polytopes from their 1-skeleta 234b
Reconstruction of duals of cubical zonotopes from their 1-skeleta 234b
Reconstruction of polytopes from their (d-2)-skeleta 234a
Reconstruction of simple polytopes from their 1-skeleta 234a
Reconstruction of simplicial polytopes from their [d/2]-skeleta 234a
Reconstruction of zonotopes from their 1-skeleta 234b
Reducible convex set 26 322
Reducible polytope 322
Reduction, proof of Steinitz’s theorem 237 (see also “ -transformation”)
Reduction, proof on realizable sequences 272
Refinement map 199
Refinement of a complete graph in a polytopal graph 214
Refinement of a complex 199
Refinement of the boundary complex of a simplex 219
Reflection group see “Finite reflection group”
Regular cell complex, Eulerian lattices 142b
Regular cell complex, intersection property 142b
Regular cell complex, strongly 142b
Regular polytope 423a
Regular polytope, affinely 412
Regular polytope, combinatorially 413
Regular polytope, projectively 412
Regular-faced polytope 414
Related polytopes 50 52c
Relative boundary 9
Relative interior 9 428a
Representation of a polytope 52a
Representation of a polytope as a section 96a
Representation of a polytope, -description 52a
Representation of a polytope, -description 52a
Representation of a polytope, alternative 52b
Representation of a polytope, coding size 52a 296d
Representation of a polytope, computation 52a
Representation of a polytope, oracle 52b
Representation of a polytope, polynomial inequalities 52b
Reverse search 52a
Rigidity theorem 411
Rigidity theory, lower bound theorem 198a
Rigidity theory, Steinitz’s theorem 296a
Rotation distances of trees 30b
Rubber band method 296a
Scalar product 2
Scheme of a polytope 90
Scheme, abstract 91
Schlegel diagram 43 52c
Schlegel diagram, 3-polytopes 235 244
Schlegel diagram, circumcircles 287
Schlegel diagram, embedding graphs into cyclic 4-polytopes 212
Schlegel diagram, history 127
Schlegel diagrams, 3-diagrams not being 219
Section of a polytope 71
Selection theorem 10 325
Self-dual polytope 48
Self-dual polytope, pyramids 69
Self-dual polytopes, density of the family of 3-dimensional 82
Self-dual polytopes, enumeration of 3-dimensional 289
Semi-algebraic set 52b
Semi-algebraic variety 96a
Semiregular polytope 423a
Semispaces 13
Separated set 10
Separated sets, strictly 10
Separation in graphs 217
Sewing 129a
Shadow vertex pivot rule 389b
Shape matching 315b
Shellability of polytopes 142a
Shellability, upper bound theorem 198a
Simple circuit 356 381
Simple complex 206
Simple d-arrangement 391
Simple path 356
Simple polytope 58
Simple polytope, affine hull of f-vectors 170
Simple polytope, decomposability 321
Simple polytope, degree of total separability 218
Simple polytope, h-simple polytope 58
Simple polytope, incidence equation 144
Simple polytope, Perles’ conjecture on the facet subgraphs 234a
Simple polytope, reconstruction from 1-skeleton 234a
Simple polytope, refinements of boundary complexes 206
Simple polytope, relations between the Steiner points of the faces 311
simplex 53
Simplex -path 376
Simplex -path, unambiguous 379
Simplex algorithm 377 389b
Simplex height 376
Simplex, combinatorial type 53
Simplex, embedding skeleta of simplices 210
Simplex, generalized bipyramid of two simplices 64
Simplex, sections of simplices 71
Simplex, simplices as regular polytopes 412
Simplex, spherical 306
Simplicial complex 59
Simplicial complex, embedding 67 202 224a
Simplicial complex, minimal face numbers 179
Simplicial polyhedral complex 51
Simplicial polytope 57
Simplicial polytope with d+2 vertices 97
Simplicial polytope, angle-sums relations 307
Simplicial polytope, Dehn — Sommerville equations 171a
Simplicial polytope, f-vector and h-vector 171a
Simplicial polytope, f-vectors 145
Simplicial polytope, g-theorem 171b
Simplicial polytope, Gale-transforms 88
Simplicial polytope, h-vector and face ring 171a
Simplicial polytope, k-simplicial polytope 58
Simplicial polytope, neighborly polytopes 124
Simplicial polytope, reconstruction from [d/2]-skeleton 234a
Simplicial polytope, stability 69
Simplicial polytope, upper bound problem 172
Simplicial polytopes, density of the family of 81
Simplicial polytopes, number of see “Number”
Simplicial sphere, centrally symmetric see “Centrally symmetric”
Simplicial sphere, g-theorem 198b
Simplicial sphere, polytopality 121a
Simplicial sphere, upper bound theorem 171a 198a
Simplicial spheres, number of see “Number”
Singular of degree s 269
Skeleton 138
Skeleton of a simplex 201 210
Skeleton, k-skeleton 138
Slack variable 96a
Snake-in-the-box code 382
Space-filling 411
Spanning tree 296
Sphere 5
Sphere, shellable 96b
Sphere, simplicial see “Simplicial sphere”
Spheres, convexity on 10 30
Spherical arrangement 409
Spherical polytope, arrangements 392
Spherical polytope, Euler’s equation 142
Spherical polytope, incidence equations 145
Spherical simplex 306
Spherical space 7a
Spherically convex set 10 30
SPREAD 382
Stable equivalence 96a
Stacked polytope 198b
Standard Gale-diagram 109
Stanley — Reisner ring 198a 198c
Star 40
Star, k-star 138 301
Star-diagram 103
Star-diagram, simplicial polytopes 114
Steep -path 376
Steep height 376
Steepest edge pivot rule 389b
Steiner point 308 315a
Steiner point, history 313
Steiner point, translation classes 317
Steiner point, valuation property 315
Steinitz’s theorem on 3-polytopes 235 296a—296b
Steinitz’s theorem on 3-polytopes, analogue for 2-arrangements 409
Steinitz’s theorem on 3-polytopes, enumeration of 3-polytopes 92 290
Steinitz’s theorem on 3-polytopes, modifications 244
Steinitz’s theorem on 3-polytopes, projective uniqueness 68
Steinitz’s theorem on convex hulls 17
Straszewicz’ theorem 19
Stretchable arrangement 408
Strict -path 375
Strict height 376
Strictly antipodal subset of 128 129b
Strong d-ambiguity 225 228
Strong monotone Hirsch conjecture 355b
Strongly isomorphic polytopes 52c
Strongly regular cell complex see “Regular cell complex”
Subdivisions of complete graphs 224b
subspace 3
Subspace arrangements 410b
Summand of a polytope 316
Supporting function 13 326
Supporting hyperplane 10 13
Sweep 142a
Sylvester — Gallai theorem 410b
Sylvester’s problem 404 (see also “Sylvester — Gallai theorem”)
Symmetry of a polytope 120
Symmetry of a polytope, central see “Centrally symmetric polytope”
Symmetry of a polytope, regular polytopes 412
Tarski’s procedure 96b
Tarski’s theorem 91
Threshold gate 121b
Tile type 423a
Tiling of 423a
Topological complex, Euler characteristic 142
Topological complex, incidence equation 145
Topological complex, polyhedral, convex, geometric complex 39
Toric g-vector 198c
Toric h-vector 171b
Toric variety 52c
Toric variety and h-vector 171b
Toric variety and the g-theorem 171b
Toric variety, cohomology 198c
Toric variety, Morse function 142a
Torus 207 253
Totally positive matrix 129b
Totally separated 217
Totally separated, degree of total separability 217
Tower 37
Towers of Hanoi 383
Transformation, adjoint 50
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