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 Call number: SPRINGER-2003-9781461300199:ONLINE Show nearby items on shelf Title: Convex Polytopes Author(s): Branko Grünbaum Date: 2003 Edition: Second Edition Size: 1 online resource (471 p.) Note: 10.1007/978-1-4613-0019-9 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!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaadG % aGmUaaaeacaYOaiaiJigdaaeacaYOaiaiJikdaaaacbiGaiaiJ-rga % aiaawUfacaGLDbaaaaa!40CC! $$\left[ {\frac{1} {2}d} \right]$$-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 ISBN: 9781461300199 Series: eBooks Series: SpringerLink (Online service) Series: Springer eBooks Series: Graduate Texts in Mathematics: 221 Keywords: Mathematics , Convex geometry , Discrete geometry , Mathematics , Convex and Discrete Geometry Availability: Click here to see Library holdings or inquire at Circ Desk (x3401) Click to reserve this book Be sure to include your ID please. 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