  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.
