Polyhedral Gauss–Bonnet theorems and valuations
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.
KeywordsGauss–Bonnet theorem Polyhedron Polyhedral curvature Valuation Critical point theorem
Mathematics Subject Classification52B05 52B45 52B70
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