Polyhedral Gauss–Bonnet theorems and valuations

Original Paper


The Gauss–Bonnet theorem for a polyhedron (a union of finitely many compact convex polytopes) in n-dimensional Euclidean space expresses the Euler characteristic of the polyhedron as a sum of certain curvatures, which are different from zero only at the vertices of the polyhedron. This note suggests a generalization of these polyhedral vertex curvatures, based on valuations, and thus obtains a variety of polyhedral Gauss–Bonnet theorems.


Gauss–Bonnet theorem Polyhedron Polyhedral curvature Valuation Critical point theorem 

Mathematics Subject Classification

52B05 52B45 52B70 


  1. Akopyan, A., Bárány, I., Robins, S.: Algebraic vertices of non-convex polyhedra. arXiv:1508.07594v2 (2017)
  2. Amelunxen, D., Lotz, M.: Intrinsic volumes of polyhedral cones: a combinatorial perspective. Discrete Comput. Geom. 58, 371–409 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. Banchoff, T.: Critical points and curvature for embedded polyhedra. J. Differ. Geom. 1, 245–256 (1967)MathSciNetCrossRefMATHGoogle Scholar
  4. Bloch, E.D.: The angle defect for arbitrary polyhedra. Beitr. Algebra Geom. 39, 379–393 (1998)MathSciNetMATHGoogle Scholar
  5. Brehm, U., Kühnel, W.: Smooth approximation of polyhedral surfaces regarding curvatures. Geom. Dedicata 12, 435–461 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. Brin, I.A.: Gauss–Bonnet theorems for polyhedra (in Russian). Uspekhi Mat. Nauk 3, 226–227 (1948)Google Scholar
  7. Budach, L.: Lipschitz–Killing curvatures of angular partially ordered sets. Adv. Math. 78, 140–167 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. Cheeger, J., Müller, W., Schrader, R.: On the curvature of piecewise flat spaces. Commun. Math. Phys. 92, 405–454 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. Chen, B.: The Gram-Sommerville and Gauss-Bonnet theorems and combinatorial geometric measures for noncompact polyhedra. Adv. Math. 91, 269–291 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. Chen, B.: The incidence algebra of polyhedra over the Minkowski algebra. Adv. Math. 118, 337–365 (1996)MathSciNetCrossRefMATHGoogle Scholar
  11. Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)MathSciNetCrossRefMATHGoogle Scholar
  12. Grünbaum, B., Shephard, G.C.: Descartes’ theorem in \(n\) dimensions. L’Enseignement Math. 37, 11–15 (1991)MathSciNetMATHGoogle Scholar
  13. Hadwiger, H.: Eckenkrümmung beliebiger kompakter euklidischer Polyeder und Charakteristik von Euler-Poincaré. L’Enseignement Math. 15, 147–151 (1969)MATHGoogle Scholar
  14. Klaus, S.: On combinatorial Gauss–Bonnet Theorem for general Euclidean simplicial complexes. Front. Math. China 11, 1345–1362 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. McMullen, P.: Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc. 78, 247–261 (1975)MathSciNetCrossRefMATHGoogle Scholar
  16. McMullen, P.: Review of Budach (1989) in Mathematical Reviews, MR1029089 (1991)Google Scholar
  17. Morvan, J.-M.: Generalized Curvatures. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  18. Schneider, R.: Ein kombinatorisches Analogon zum Satz von Gauss–Bonnet. Elem. Math. 32, 105–108 (1977)MathSciNetMATHGoogle Scholar
  19. Schneider, R.: Kritische Punkte und Krümmung für die Mengen des Konvexringes. L’Enseignement Math. 23, 1–6 (1977)MATHGoogle Scholar
  20. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)Google Scholar

Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-UniversitätFreiburg i. Br.Germany

Personalised recommendations