Skip to main content Accessibility help

Angle-sum relations for polyhedral sets

  • Peter McMullen (a1)


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.



Hide All
1.Brianchon, C. J.. Théorème nouveau sur les polyèdres. J. Ecole (Royale) Polytechnique, 15 (1837), 317319.
2.Gram, J. P.. Om Rumvinklerne i et Polyeder. Tidsskr. Math. (Copenhagen) (3), 4 (1874), 161163.
3.Grünbaum, B.. Convex Polytopes (Wiley-Interscience, London, 1967).
4.Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957).
5.Jessen, B. and Thorup, A.. The algebra of polytopes in affine spaces. Math. Scand., 43 (1978), 211240.
6.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc, 78 (1975), 247261.
7.McMullen, P.. Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3), 35 (1977), 113135.
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), 170247.
9.Rockafellar, R. T.. Convex Analysis (Princeton, 1968).
10.Rota, G.-C.. On the foundations of combinatorial theory, I: Theory of Möbius functions. Z Wahrscheinlichkeitstheorie verw. Geb., 2 (1964), 340368.
11.Sah, C.-H.. Hilbert's third problem: Scissors congruence (Pitman, San Francisco, 1979).
12.Sah, C.-H.. Scissors congruences, I: The Gauss-Bonnet map. Math. Scand., 49 (1981), 181210.
13.Sallee, G. T.. Polytopes, valuations, and the Euler relation. Canad. J. Math., 20 (1968),: 14121424.
14.Shephard, G. C.. An elementary proof of Gram's theorem for convex polytopes. Canad. J. Math., 19 (1967), 12141217.
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), 103119.
16.Tverberg, H.. How to cut a convex polytope into simplices. Geom. Ded., 3 (1974), 239240.
MathJax is a JavaScript display engine for mathematics. For more information see


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed