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. Language: English