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Grünbaum B. — Convex Polytopes
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Название: Convex PolytopesАвтор: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: Second EditionГод издания: 2003Количество страниц: 466Добавлена в каталог: 30.06.2008Операции: Положить на полку |
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Cubical polytope, angle-sums relations 313 Cubical polytope, f-vectors 156 Cubical polytope, h-vector 171b Cubical polytope, neighborly 69b Cubical polytopes, family of 82 Cuboctahedron 340b 423a cuboid 59 Cuboid, face numbers 156 Cuboid, images of cubes 66 Cummings criterion 294 394 Curvature centroid 307 314 Cut polytope 69a Cutting plane approach 69a Cyclic oriented matroid 7b Cyclic polytope 61 Cyclic polytope with d+2 vertices 69b 98 168 Cyclic polytope, affine hull of f-vectors 147 Cyclic polytope, ambiguity of complexes 225 Cyclic polytope, embedding graphs into 212 Cyclic polytope, extremal path-lengths 368 Cyclic polytope, f-vectors of 4-polytopes 191 Cyclic polytope, height of the duals 389b Cyclic polytope, Hirsch conjecture for the duals 355b Cyclic polytope, history 127 Cyclic polytope, neighborly poly topes 122 124 Cyclic polytope, partial proof of the upper bound theorem 106 175 Cyclic polytope, realizations 129b Cyclic polytope, realizations with full symmetry 69b Cyclic polytope, spanning trees of bounded valence 296 Cyclic polytope, totally positive matrices 129b Cyclic polytope, twisted lexicographic order 389a Cyclic polytopes, edges of 4-dimensional 66 Cyclically k-connected 3 polytope 365 Cylindrical algebraic decomposition 96b d-realizability of abstract d-complexes 206 d-realizability of abstract schemes 91 d-realizability of lattices 231 d-realizability of star-diagrams 106 114 d-step conjecture 349 355a d-step conjecture, 354 d-unambiguity of (d-2)-skeleta of d-poly topes 228 d-unambiguity of 1-skeleta of capped cubical d-polytopes 234b d-unambiguity of 1-skeleta of duals of cubical d-zonotopes 234b d-unambiguity of 1-skeleta of simple polytopes 234a d-unambiguity of 1-skeleta of zonotopes 234b d-unambiguity of [d/2]-skeleta of d-cubes 234b d-unambiguity of [d/2]-skeleta of simplicial d-polytopes 234a Dantzig figure 350 Dantzig’s pivot rule 389b Decomposable polytope 318 Decomposable polytope, Blaschke addition 333 Decomposition and projective transformations 340a Decomposition of 3-polytopes 340a Decomposition of 4-poly topes 340a Decomposition of general convex sets 340a Decomposition of polyhedra 315b 340a Decomposition of simple polytopes 340a Deformed product 389b Degree of singularity 269 Degree of total separability 217 Dehn — Sommerville equations for Eulerian manifolds 152 Dehn — Sommerville equations for simple arrangements 392 Dehn — Sommerville equations for simplicial 3-polytopes 189 Dehn — Sommerville equations for simplicial polytopes 146 171a Dehn — Sommerville equations, cyclic polytopes 168 Dehn — Sommerville equations, equivariant generalization 171b Dehn — Sommerville equations, generalized 198c Dehn — Sommerville equations, history 170 Dense family of polytopes 81 Description of a polytope see “Representation of a polytope” Design of experiments 340b Diagram 44 Diagram, Blaschke 417 Diagram, central 121b Diagram, dual 45 Diagram, dualizable 52c Diagram, edge-facet (ef-) 350 Diagram, Gale see “Gale-diagram” Diagram, inverted 45 Diagram, invertible 52c Diagram, non-polytopal 52c 224a Diagram, Schlegel 43 (see also “Schlegel diagram”) Diagram, simple 52c Diagram, standard see “Standard Gale-diagram” Diagram, star see “Star-diagram” Diagrams, 2-diagrams 244 286 Diagrams, 3-diagrams, that are not Schlegel 188 219 Diameter 419 Diameter of a Gale-diagram 109 Diameter of a polytope 341 Diameter of a set 5 Diameter of duals of neighborly polytopes 355b Diameter, computation 423c Diameter, facet-diameter 343 Diameter, upper bounds 355a—355b Difference set of a convex set 322 Difference set of a simplex 340b DIMENSION 3 Dimensional ambiguity 225 229 Directed diameter see “Diameter” Discrete code 382 Dissection 315b Distance between sets 6 Distance in a metric space 5 Distance, Hausdorff 6 Distance, path-distance 341 Dodecahedron, 3-polytopes without triangles and quadrilaterals 198d Dodecahedron, distances between vertices 420 Double counting 143 Double-description method 52b Dual polytope 46 Dual polytope, self see “Self-dual polytope” Eberhard’s Theorem 253 Eberhard’s theorem, extensions 296b 405 Eberhard’s theorem, history 291 EDGE 31 200 Edge-facet (ef-) diagram 350 Elementary transformation 236 Ellipsoid method 30b Embeddings of simplicial complexes 224a Entangled 210 Enumeration of 2-arrangements 394 Enumeration of 3-polytopes 296c Enumeration of 4-polytopes with 8 vertices 96b Enumeration of combinatorial types of polytopes 90 Enumeration of neighborly 4-polytopes with 10 vertices 96b Enumeration of neighborly 6-polytopes with 10 vertices 96b Enumeration of self-dual 3-polytopes 52d Enumeration of simplicial 3-polytopes 296c Enumeration of simplicial 4-polytopes with 9 vertices 96b Enumeration of the faces of a polytope 90 Equifacetted manifold 423a Equifacetted polytope 423a Equilateral set 423d Equilibrated system 332 Equilibrated system of vectors 340c Equilibrated system, fully 332 Equilibrium embedding 296a Equisurrounded 3-polytopes 292 EQUIVALENCE 38 Equivalence of arrangements 394 Equivalence, k-equivalence 225 231 Equivalent vertices 413 Equivalent, combinatorially 38 Equivalent, projectively see “Projectively equivalent” Equivariant topology 30b Erdoes — Szekeres theorem 30b Euclidean space 7a Euler characteristic 138 Euler hyperplane 136 Euler hyperplane, cubical polytopes 156 Euler hyperplane, quasi-simplicial polytopes 153 Euler — Poincare formula see “Euler’s equation” Eulerian lattice 142b Eulerian manifold 141 Eulerian manifold, boundary complexes of polytopes 141 Eulerian manifold, Dehn — Sommerville equations 152 Eulerian manifold, upper bound conjecture 182 Eulerian poset 142b Euler’s equation 131 Euler’s equation for 3-polytopes 236 Euler’s equation for arrangements 391 Euler’s equation, angle-sums 297 Euler’s equation, Dehn — Sommerville equations 145 Euler’s equation, generalization 137 Euler’s equation, Gram’s equation 315a Euler’s equation, sweeps 142a Euler’s theorem 131 291 Exposed point 17 30a Exposed point, k-exposed point 20 External angle 308 Extreme point 17 30a Extreme point, k-extreme point 20 f-vector 130 f-vector, 4-polytopes 191 198d f-vector, arrangements 390 399 f-vector, cubical polytopes 156 f-vector, Eulerian manifolds 152 f-vector, Eulerian posets 142b f-vector, quasi-simplicial polytopes 153 f-vector, simplicial polytopes see “Simplicial polytope” Face function 30a Face lattice, Euler’s equation 142b Face lattice, reconstruction from an oracle 52b Face of a convex set 17 Face of a point set 88 Face of a polytope 31 Face of an arrangement 390 Face ring 198a (see also “Stanley — Reisner ring”) Face ring and h-vector 171a Face, improper 17 Face, j-face 21 Face, x-face 350 Facet 342 Facet complexity 52a Facet of a polyhedral set 26 Facet of a polytope 31 71 Facet splitting 129a Facet tile 423b Facet-diameter 342 Fan 52c Fan, centrally symmetric see “Centrally symmetric” Fan, g-theorem 198b Farkas lemma 96a Fatness of a 4-polytope 198d Feasible region 347 Fibonacci numbers 198c Final polynomial 96b Finite reflection group 171b 410b First-order sentence 91 Fixing system 423 423c Fixing system, primitive 423 Flag f-vector 198a 198c Flag h-vector 198c FLAGTOOL 224b Flat 3 Four-color problem 127 284 418 Four-color problem, cyclically 5-connected 3-polytopes 365 Four-color problem, Hamiltonian circuits 356 Four-color problem, solution 389a Fourier — Motzkin elimination method 52b Free sum 69b Fullerene 296b Fully equilibrated system 332 Functional analysis 30a 96a g-theorem 198a g-theorem, generalizations 198b g-theorem, matrix formulation 198a g-theorem, necessity part and toric varieties 171b g-theorem, polytope algebra 340a Gale-diagram 89 Gale-diagram, affine 96a Gale-diagram, construction of a non-rational 8-polytope 95 Gale-diagram, contracted 109 120 Gale-diagram, distended 109 Gale-diagram, interpretations 96a Gale-diagram, k-neighborly polytopes 126 Gale-diagram, neighborly polytopes 129b Gale-diagram, polytopes with few vertices 108 Gale-diagram, quotients of neighborly polytopes 129b Gale-diagram, standard 109 Gale-diagram, winding numbers 198b Gale-transform 86 119 Gale’s evenness condition 62 Gale’s evenness condition, star-diagrams 106 Gale’s lemma 129b General position 87 126 General position of hyperplanes 58 General position of points 4 57 Generalized arrangement 407 Generalized bipyramid 69b Generalized Dehn — Sommerville equations 198c Generalized lower bound conjecture 198b Generalized lower bound theorem 198b Generalized prismoid 65 Generalized regular polytope 423a Geodesic arc 239 Geometric cell complex 51 Geometry of numbers 30a 52a Gosset — Elte polytope 171b Gradient, nonbasic 378 Gram’s equation 298 304 315a Graph 212 Graph of a complex 138 Graph of a polytope 212 341 Graph of an arrangement 397 Graph, planar 235 358 Gray code 382 389b Greatest increase pivot rule 389b h-vector 171a h-vector of a cubical polytope 171b h-vector of a simplicial polytope see “Simplicial polytope” Hadamard’s conjecture 423d Hadwiger — Nelson problem 423b Half cube 171b Halfspace 2 Halfspace, polytopes 31 Halfspaces, intersections of 12 Hamilton game 356 Hamiltonian circuit 356 368 Hamiltonian path 356 366 Hamiltonicity, four-color problem 389a Hamiltonicity, graphs of simple polytopes 389a Hanner polytope 224b Hard Lefschetz theorem 171b Hausdorff distance 6 Hausdorff measures of skeletons 423b Hausdorff metric, approximation of polytopes by vector sums 324 Hausdorff metric, continuity of 416 Hausdorff metric, lower semicontinuity of 83 Height of an element in a poset 231 Height, simplex 376 Height, steep 376 Height, strict 376 Helly-type theorem 22 30b 126 Hexagon 340b Hilbert, tenth problem 96b Hilbert, third problem 315b Hirsch conjecture 349 355a Hirsch conjecture for simplicial spheres 355b
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