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The maximum numbers of faces of a convex polytope

Published online by Cambridge University Press:  26 February 2010

P. McMullen
P. McMullen
Affiliation:
Western Washington State College, Bellingham, WashingtonandUniversity College, London

Abstract

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In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 179 - 184
Copyright
Copyright © University College London 1970

References

Brückner, M. 1893, “Die Elemente der vierdimensionalen Geometrie mit besonderer Berücksichtigung der Polytope”, Jber. Ver. Naturk. Zwickau (1893).Google Scholar
Bruggesser, H. and Mani, P., 1970, “Shellable decompositions of cells and spheres”, to be published.CrossRefGoogle Scholar
Dehn, M. 1905, “Die Eulersche Formel in Zusammenhang mit dem Inhalt in der nicht-Euklidischen Geometrie”, Math. Ann., 61 (1905), 561–586.CrossRefGoogle Scholar
Fieldhouse, M. 1961, Linear programming, Ph.D. Thesis, Cambridge Univ. (1961). (Reviewed in Operations Res., 10 (1962), 740.)Google Scholar
Gale, D. 1964, “On the number of faces of a convex polytope”, Canad. J. Math., 16 (1964), 12–17.Google Scholar
Grunbaum, B. 1967, Convex polytopes (John Wiley and Sons, London–New York–Sydney, 1967).Google Scholar
Grunbaum, B. 1970, Polytopes, graphs and complexes, to be published.CrossRefGoogle Scholar
Klee, V. L., 1964a, “A combinatorial analogue of Poincare's duality theorem”, Canad. J. Math., 16 (1964), 517–531.Google Scholar
Klee, V. L., 1964b, “The number of vertices of a convex polytope”, Canad. J. Math., 16 (1964), 701–720.CrossRefGoogle Scholar
McMullen, P. 1970a, “On a problem of Klee concerning convex polytopes”, Israel J. Math., 8 (1970), 1–4.CrossRefGoogle Scholar
McMullen, P. 1970b, “On the upper-bound conjecture for convex polytopes”, J. Combinatorial Theory, to be published.CrossRefGoogle Scholar
McMullen, P. and Shephard, G. C., 1970, Convex polytopes and the upper-bound conjecture, London Math. Soc. Lecture Notes Series, Volume 3, to be published.Google Scholar
McMullen, P. and Walkup, D. W., 1970, “A generalized lower-bound conjecture for simplicial polytopes,” to be published.CrossRefGoogle Scholar
Motzkin, T. S., 1957, “Comonotone curves and polyhedra”, Bull. Amer. Math. Soc., 63 (1957), 35.Google Scholar
Sommerville, D. M. Y., 1927, “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