Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T08:51:38.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Angle-sum relations for polyhedral sets

Published online by Cambridge University Press:  26 February 2010

Peter McMullen
Peter McMullen
Affiliation:
Department of Mathematics, University College, Gower Street, London. WC1E 6BT
Get access

Abstract

The Brianchon-Gram and Sommerville theorems on angle-sums for convex polytopes and polyhedral cones are here shown to be particular cases of an angle-sum relation for general polyhedral sets. The new relation is proved on the level of an equidissectability theorem, and this approach yields yet other angle-sum relations, including a different generalization of the Brianchon-Gram theorem. Further results extend, again to equidissections, earlier angle-sum relations of the author and others.

Type
Research Article
Information
Mathematika , Volume 33 , Issue 2 , December 1986 , pp. 173 - 188
Copyright
Copyright © University College London 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Brianchon, C. J.. Théorème nouveau sur les polyèdres. J. Ecole (Royale) Polytechnique, 15 (1837), 317–319.Google Scholar
2.Gram, J. P.. Om Rumvinklerne i et Polyeder. Tidsskr. Math. (Copenhagen) (3), 4 (1874), 161–163.Google Scholar
3.Grünbaum, B.. Convex Polytopes (Wiley-Interscience, London, 1967).Google Scholar
4.Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957).CrossRefGoogle Scholar
5.Jessen, B. and Thorup, A.. The algebra of polytopes in affine spaces. Math. Scand., 43 (1978), 211–240.CrossRefGoogle Scholar
6.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc, 78 (1975), 247–261.CrossRefGoogle Scholar
7.McMullen, P.. Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3), 35 (1977), 113–135.CrossRefGoogle Scholar
8.McMullen, P. and Schneider, R.. Valuations on convex bodies. In Convexity and its Applications, ed. Gruber, P. M. and Wills, J. M. (Birkhäuser, Basel, 1983), 170–247.CrossRefGoogle Scholar
9.Rockafellar, R. T.. Convex Analysis (Princeton, 1968).Google Scholar
10.Rota, G.-C.. On the foundations of combinatorial theory, I: Theory of Möbius functions. Z Wahrscheinlichkeitstheorie verw. Geb., 2 (1964), 340–368.CrossRefGoogle Scholar
11.Sah, C.-H.. Hilbert's third problem: Scissors congruence (Pitman, San Francisco, 1979).Google Scholar
12.Sah, C.-H.. Scissors congruences, I: The Gauss-Bonnet map. Math. Scand., 49 (1981), 181–210.CrossRefGoogle Scholar
13.Sallee, G. T.. Polytopes, valuations, and the Euler relation. Canad. J. Math., 20 (1968),: 1412–1424.CrossRefGoogle Scholar
14.Shephard, G. C.. An elementary proof of Gram's theorem for convex polytopes. Canad. J. Math., 19 (1967), 1214–1217.CrossRefGoogle Scholar
15.Y, D. M.. Sommerville. The relations connecting the angle-sum and volume of a polytope in space of n dimensions. Proc. Roy. Soc. London, A, 115 (1927), 103–119.Google Scholar
16.Tverberg, H.. How to cut a convex polytope into simplices. Geom. Ded., 3 (1974), 239–240.CrossRefGoogle Scholar