Geometriae Dedicata

, Volume 68, Issue 1, pp 21–28 | Cite as

f-Vectors of Polyhedra

Article

Abstract

The f-vector of a triangulation of a polyhedron X is the numbers of simplices at various dimensions. We prove that the affine span of f-vectors of X has dimension (n+s+1)/2, where n is the dimension of X, and s is the dimension of the part of X that is singular with respect to the local Euler characteristic.

polyhedron f-vector affine span Euler characteristic link. 

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References

  1. [BaL]
    Bayer, M. and Lee, C.: Combinatorial aspects of convex polytopes. In: P. M. Gruber and J. M. Wills (eds), Handbook of Convex Geometry, Elsevier Science, 1993, pp. 485-534Google Scholar
  2. [BiL]
    Billera, L. J. and Lee, C.: A proof of the sufficiency ofMcMullen's conditions for f-vectors of simplicial convex polytopes, J. Combin. Theory A 31 (1981), 237-255.Google Scholar
  3. [Br]
    Brøndsted, A.: An Introduction to Convex Polytopes, Graduate Texts inMath. 90, Springer-Verlag, Berlin, Heidelberg, New York, 1983.Google Scholar
  4. [CY1]
    Chen, B. F. and Yan, M.: Linear conditions on the number of faces of manifolds with boundary, Adv. Appl. Math. 19 (1997), 144-168.Google Scholar
  5. [CY2]
    Chen, B. F. and Yan, M.: Eulerian 2-strata spaces, Preprint, 1996.Google Scholar
  6. [G1]
    Grünbaum, B.: Convex Polytopes, Pure Appl. Math. 16, Wiley-Interscience, New York, 1967.Google Scholar
  7. [G2]
    Grünbaum, B.: Polytopes, graphs, and complexes, Bull. Amer.Math. Soc. 76 (1970), 1131-1201.Google Scholar
  8. [JR]
    Jungerman, M. and Ringel, G.: Minimal triangulations on orientable surfaces, Acta Math. 145 (1980), 121-154.Google Scholar
  9. [Ka]
    Katona, G.: A theorem of finite sets, In: P. Erdös and G. Katona (eds), Theory of Graphs, Proceeding of Tihany Conference, 1966, Academic Press, New York, Akadèmia Kiadò, Budapest, 1968, pp. 187-207.Google Scholar
  10. [Kl]
    Klee, V.: A combinatorial analogue of Poincarè's duality theorem, Canad. J. Math. 16 (1964), 517-531.Google Scholar
  11. [Kr]
    Kruskal, J. B.: The number of simplices in a complex, In: Mathematical Optimization Techniques, University of California Press, 1963, 251-278.Google Scholar
  12. [M]
    McMullen, P.: On the upper bound conjecture for convex polytopes, J. Combin. Theory B 10 (1971), 187-200.Google Scholar
  13. [MS]
    McMullen, P. and Shephard, G. C.: Convex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lecture Note Ser. 3, Cambridge University Press, London, 1971.Google Scholar
  14. [R]
    Ringel, G.: DeWie man die geschlossenen nichtorientierbaren Flächen in möglichst wenig Dreiecke zerlegen kann, Math. Ann. 130 (1955), 317-326.Google Scholar
  15. [RS]
    Rourke, C. P. and Sanderson, B. J.: Introduction to Piecewise-Linear Topology, Springer-Verlag, Berlin, Heidelberg, New York, 1972.Google Scholar
  16. [So]
    Sommerville, D. M. Y.: The relations connecting the angle-sums and volume of a polytope in space of n-dimensions, Proc. Roy. Soc. London Ser. A 115 (1927), 103-119.Google Scholar
  17. [St]
    Stanley, R. P.: On the number of faces of simplicial convex polytopes, Adv. in Math. 35 (1980), 236-238.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Min Yan
    • 1
  1. 1.Department of MathematicsHong Kong University of Science and TechnologyClear Water BayHong Kong

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