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Grünbaum B. Convex Polytopes
Grünbaum B.  Convex Polytopes


: Convex Polytopes

: Grünbaum B.


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.

: en

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ed2k: ed2k stats

: Second Edition

: 2003

: 466

: 30.06.2008

: | | ID
$(\mathcal{P})$-realizable complex      see $(\mathcal{P}^d)$-realizable
$(\mathcal{P}^d)$-realizable      199
$(\mathcal{P}^d)$-realizable, ambiguously, complexes      225
$(\mathcal{P}^d)$-realizable, dimensionally unambiguously, complexes for d=5      228
$(\mathcal{P}^d)$-realizable, graphs      213 217
$(\mathcal{P}^d)$-realizable, graphs for d=3      235
$W_{\nu}$ path      354
$W_{\nu}$-conjecture      354 355a
$W_{\nu}$-conjecture for simplicial spheres      355b
$\Delta Y$-transformation      296a
$\mathcal{H}$-description      see Representation of a polytope
$\mathcal{V}$-description      see Representation of a polytope
$\zeta$-vector      417
0/1-poly tope      69a
0/1-poly tope, Borsuks conjecture      423b
0/1-poly tope, Hirsch conjecture      355a
0/1-poly tope, maximal number of facets      69a
0/1-poly tope, random      69a
0/1-poly tope, simple      340a
120-cell      423b
24-cell      69a 423b
24-cell, f-vectors of 4-polytopes of type (2,2)      169
3-diagram      see Diagram
3-polytope without triangles and quadrangles      198d
3-polytope, automorphisms and symmetries      296a
3-polytope, cutting off vertices      270 284 286
3-polytope, decompositions      340a
3-polytope, directed version of Steinitzs theorem      296b
3-polytope, Eberhards theorem      296b
3-polytope, face vector      296b
3-polytope, Hamiltonicity      389a
3-polytope, inscribed in a sphere      296c
3-polytope, order dimension of the face lattice      296d
3-polytope, prescribing a shadow boundary      296b
3-polytope, prescribing the shape of a facet      296b
3-polytope, realization space      296b
3-polytope, realization with edges tangent to a sphere      296a
3-polytope, self-dual      52d
3-polytope, size of coordinates      296b
3-polytope, Steinitzs theorem      296a
3-polytope, vertex vector      296b
3-polytopes, enumeration of      see Enumeration
3-polytopes, number of      see Number
3-realizable sequence      253 (see also Eberhards theorem)
3-realizable sequence, with $p_{i}=0$ for $i\geq7$      271
4-polytope, centrally symmetric, with 12 vertices      121b
4-polytope, coding sizes of coordinates      296d
4-polytope, complexity      198d
4-polytope, decompositions      340a
4-polytope, f-vectors      198d
4-polytope, fatness      198d
4-polytope, flag f-vectors      198c
4-polytope, Hamiltonicity      389a
4-polytope, non-rational      96a
4-polytope, universality theorem      96a
4-polytope, visualization      52c
4-polytopes, enumeration of      see Enumeration
4-polytopes, number of      see Number
4-realizable sequence      282
5-realizable sequence      271
600-cell      423b
Abstract complex      206
Abstract scheme      91
Addition, Blaschke      see Blaschke sum
Addition, Minkowski      316
Addition, vector      9 38 316
Adjoining polytopes      83
Adjoint transformation      50 75
Admissible pair of numbers      352
Affine automorphism      77
Affine combination      2
Affine convex geometry      30b
Affine dependence      2 85
Affine equivalence      5 74 89
Affine Gale-diagram      96a
Affine hull      3
Affine transformation      4 9
Affine transformation, piecewise      41
Affine variety      3
Affinely regular polytope      412
Algebraic geometry      198c
Algebraic numbers      96b
Algebraic shifting      198c
Algebraic shifting, generalized lower bound theorem      198b
Algebraic variety, topological complexity      121a
Algebraic variety, toric      see Toric variety
Almost-neighborly set      126
Ambiguity      225
Ambiguity, dimensional      225
Ambiguity, strong d-ambiguity      225
Ambiguity, weak d-ambiguity      226
Analogous polytopes      52c
angle      297
Angle, external      308
Angle-sum      297
Angle-sums relations, cubical polytopes      313
Angle-sums relations, simplices      304
Angle-sums relations, simplicial polytopes      307
Antipodal, k-antipodal, pair      421
Antipodal, pair      420
Antipodal, polytope      420
Antipodal, strictly, subset of $R^{d}$      128
Antiprism      66 215
Antistar      40
Apex of a cone      9 23
Apex of a pyramid      54
Approximation of polytopes      326
Archimedean solid      413
Area function      339
Arrangement of pseudo-lines      408
Arrangement, equivalence      394
Arrangement, f-vectors      399
Arrangement, generalized      407
Arrangement, index of a simple 2-arrangement      394
Arrangement, simple      391
Arrangement, spherical      409
Arrangement, stretchable      408
Arrangements, nonsimple 2-arrangement      396
Associated cone      49
Balinskis theorem      213
Balinskis theorem, directed version      224a
Ball      5
Barnette sphere      224a
Barycenter      315a
Basic face of a Schlegel diagram      43
Basic variable      378
Basis of a pyramid      54
Bending facets      296a
Beneath      78
Beneath-beyond method      96b
Beyond      78
Bimatrix game      423c
Bipyramid      55
Bipyramid, r-fold      55
Bipyramids, indecomposability of r-fold      323
Bipyramids, r-fold, and pyramidoids      64
Blaschke diagram      417
Blaschke sum      333 340c
Blaschkes selection theorem      10 325
Block      255
Borsuk Ulam theorem      201 210 224a
Borsuks problem      418 423b 423c
Boundary complex      40 199
Boundary complex, Eulerian manifolds      141
Boundary complex, refinements of simplex skeleta      200
Boundary of a set      6
Boundary-free complex      50
Bounded d-step conjecture      see d-step conjecture
Bounded Hirsch conjecture      see Hirsch conjecture
Bounded set      5 23
Brueckner sphere      224a
Bruggesser Mani shelling      see Line shelling
Brunn Minkowski theorem      338
Brunn Minkowski theory      340c
Caratheodorys theorem      15
Cauchy sequence      5
Cauchys Rigidity Theorem      411
cd-index      198a 198c
CDD      52b
Cell complex      51
Cell of an arrangement      390
Center of gravity      315a
Central arrangement      410a
Centrally symmetric polytope      114
Centrally symmetric polytope with few vertices      120
Centrally symmetric polytope, 2-neighborly      121b
Centrally symmetric polytope, affine hull of f-vectors      169
Centrally symmetric polytope, Blaschke sums of parallelotopes      335
Centrally symmetric polytope, degree of total separability      218
Centrally symmetric polytope, Euler hyperplane      139
Centrally symmetric polytope, neighborly star-convex      121b
Centrally symmetric polytope, number of faces      224b
Centrally symmetric polytope, reducibility      322
Centrally symmetric polytope, simple or simplicial      198c
Centrally symmetric polytopes, 3-dimensional, and refinements of the cube      205
Centrally symmetric polytopes, 4-dimensional, with 12 vertices      121b
Centrally symmetric polytopes, d-dimensional, with 2d+2 vertices      121b
Centrally symmetric polytopes, graphs of 3-dimensional      245
Centrally symmetric polytopes, neighborly families of      129
Centrally symmetric polytopes, valences of vertices of 3-dimensional      269
Centrally symmetric, (d-1)-neighborly simplicial spheres      121b
Centrally symmetric, 2-neighborly simplicial spheres      121b
Centrally symmetric, neighborly fans      121b
Centrally symmetric, polytope      see Centrally symmetric polytope
Centroid      297 315a
Centroid, curvature      307
Characteristic cone      24
Chromatic number of $R^{d}$      423b
Circle packing theorem      296a
Circuit      200
Circuit code      381 389b
Circuit, Hamiltonian      see Hamiltonian circuit
Circuit, simple      356 381
circumcircle      286
Circumradius      423c
Circumscribable type      285
Circumsphere      284
Class of convex polytopes      325
Closed convex hull      14
Closed set      5
Closure of a set      6
Code      357
Code of spread s      382
Code, circuit      381
Code, discrete      382
Code, Gray      382
Code, path      382
Code, snake-in-the-box      382
Code, unit distance      382
Coding size of rational 4-polytopes      296d
Coding size of rational polytopes      52a
Coding size of the volume      340b
Coface of a point set      88
Cohen Macaulay property      198a
Combinatorial automorphism, 3-connected planar graphs      252
Combinatorial automorphism, number of simple 3-polytopes      289
Combinatorial automorphism, regular polytopes      413
Combinatorial automorphism, symmetries of polytopes with few vertices      120
Combinatorial complexity of arrangements      410b
Combinatorial equivalence of complexes      199
Combinatorial equivalence of diagrams      219
Combinatorial equivalence of polytopes      38
Combinatorial equivalence, isomorphism of Gale-transforms      89
Combinatorial equivalence, k-equivalence of polytopes      225
Combinatorial isomorphism      see Combinatorial equivalence
Combinatorial optimization, 0/1-polytopes      69a
Combinatorial optimization, Hirsch conjecture      355b
Combinatorial optimization, Mengers theorem      224b
Combinatorial type      38 90
Combinatorially equivalent      38 (see also Combinatorial equivalence)
Combinatorially regular polytopes      413
Commutative algebra      198c
Compact set      6
Complete metric space      6
COMPLEX      39 199
Complex hyperplane arrangement      410b
Complex of an arrangement      390
Complex, abstract      206
Complex, boundary      see Boundary complex
Complex, cell      51
Complex, linked      40
Complex, polyhedral      51
Complex, simple      206
Complex, simplicial      59 67
Complex, topological      39
Complexity of a 4-polytope      198d
Complexs, computing the number of      95
Complexs, sequences of      84
Computational convexity      52a
Computational geometry      142a 410b
Concentration of measure      30b
Cone      23
Cone, associated      49
Cone, characteristic      24
Cone, generated by a set      9
Cone, pointed      24
Cone, polyhedral      36
Cone, spanned by a set      24
Configuration      93 391
Congruent polytopes      129b
Congruent-faced polytope      414
Connected graph      212
Connected graph, k-connected graph      212
Connected sum      96b
Containment problem      423c
Content, k-content      416
Convergence      5
Convex body      see Convex set
Convex body, extremal structure      30a
Convex combination      14
Convex function      13 37
Convex hull      14
Convex hull algorithm, asymptotically optimal      52a
Convex hull algorithm, beneath and beyond      96b
Convex hull algorithm, reverse search      52 a
Convex hull problem      52a
Convex hull program      52b
Convex polyhedron      51
Convex polytope      51 (see also Polytope)
Convex set      8 (see also Convex body)
Convex set, algorithmic model      30b
Convex set, general      30a
Convex set, k-convex set      126
Convex set, protectively      29
Convex set, reducible      26
Convex set, spherically      10 30
Convex subdivision      199
Convexity, computational      52a
Convexity, generalized      30b
Convexity, hyperbolic      30b
Covering number      423c
Coxeter group      423a
Crosspolytope      see Octahedron
cube      56
Cube, codes      381
Cube, largest simplex in a      423d
Cube, polytopes without triangles      198d
Cube, sections      72
Cubical polytope      59
Cubical polytope with at most $2^{d+1}$ vertices      69b
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