Polyhedral Gauss–Bonnet theorems and valuations

  • Rolf Schneider
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 


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Copyright information

© The Managing Editors 2017

Authors and Affiliations

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

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