Convex Polytopes / by Branko Grünbaum ; edited by Volker Kaibel, Victor Klee, Günter M. Ziegler
(Graduate Texts in Mathematics ; 221)
| データ種別 | 電子ブック |
|---|---|
| 版 | Second Edition |
| 出版者 | New York, NY : Springer New York : Imprint: Springer |
| 出版年 | 2003 |
| 本文言語 | 英語 |
| 大きさ | XVI, 471 p : online resource |
書誌詳細を非表示
| 別書名 | 異なりアクセスタイトル:First Edition Prepared with the Cooperation of Victor Klee, Micha Perles, and Geoffrey C. Shephard |
|---|---|
| 内容注記 | 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 |
| 一般注記 | "The appearance of Grünbaum'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 Grünbaum 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 London) |
| 著者標目 | *Grünbaum, Branko author Kaibel, Volker editor Klee, Victor editor Ziegler, Günter M. editor SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Convex geometry LCSH:Discrete geometry FREE:Mathematics FREE:Convex and Discrete Geometry |
| 分 類 | DC23:516.1 |
| 巻冊次 | ISBN:9781461300199 
|
| ISBN | 9781461300199 |
| URL | http://dx.doi.org/10.1007/978-1-4613-0019-9 |
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