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The number of d-polytopes with d+3 vertices

Published online by Cambridge University Press:  26 February 2010

E. Keith Lloyd
E. Keith Lloyd
Affiliation:
The University, Southampton.
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Extract

A classical problem in the theory of convex polytopes is the enumeration of the distinct combinatorial types of d-polytopes with υ vertices (υ ≥ d + 1). Following Grünbaum [1] c(υ, d) will denote the number of such types. Apart from the general results c(d + l, d)= 1 and c(d + 2, d) = [¼d2], which may be established by elementary arguments, the only other known values of c(υ, d) are for small values of υ and d. These have been determined empirically; details of the most recent results are contained in [2].

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 120 - 132
Copyright
Copyright © University College London 1970

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References

1.Grünbaum, B., Convex poly topes (Interscience, London, New York and Sydney, 1967).Google Scholar
2.Grünbaum, B. and Shephard, G. C., “Convex polytopes”, Bull. London Math. Soc., 1 (1969), 257–300.CrossRefGoogle Scholar
3.McMullen, P. and Shephard, G. C., The upper bound conjecture for convex polytopes (UEA Lecture Notes, 1968).Google Scholar
4.Polya, G., “Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen”, Acta Math., 68 (1937), 145–254.CrossRefGoogle Scholar
5.Read, R. C., “Contributions to the cell growth problem”, Can. J. Math., 14 (1962), 1–20.CrossRefGoogle Scholar