Critical points and the angle defect

In a 1967 paper, Banchoff described a theory of critical points and curvature for polyhedra embedded in Euclidean space. For each convex cell complex K in R-n, and for each linear map h : R-n --> R satisfying a simple generality criterion, he defined an index for each vertex of K with respect to the map h, and showed that these indices satisfy two properties: (1) for each map h, the sum of the indices at all the vertices of K equals chi(K); and (2) for each vertex of K, the integral of the indices of the vertex with respect to all such linear maps equals the standard polyhedral notion of curvature of K at the vertex. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a more direct generalization of the angle defect. In the present paper we present an analog of Banchoff's theory that works with our generalized angle defect.