Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T00:34:03.459Z Has data issue: false hasContentIssue false
  • English
  • Français

An Elementary Proof of Gram's Theorem for Convex Polytopes

Published online by Cambridge University Press:  20 November 2018

G. C. Shephard
G. C. Shephard*
Affiliation:
University of East Anglia, England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 1214 - 1217
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Gram, J. P., Om Rumvinklerne i et Polyeder, Tidsskrift for Math. (Copenhagen) (3), 4 (1874), 161–163.Google Scholar
2. Grünbaum, B., Convex polytopes (London, New York, and Sydney, in press).Google Scholar
3. Klee, V., The Euler characteristic in combinatorial geometry, Amer. Math. Monthly, 70 (1963), 119–127.Google Scholar
4. Perles, M. A. and Shephard, G. C., Angle sums of convex polytopes, to be published in Math. Scand.Google Scholar
5. 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