THE GRAM-SOMMERVILLE AND GAUSS-BONNET THEOREMS AND COMBINATORIAL GEOMETRIC MEASURES FOR NONCOMPACT POLYHEDRA

Some non-numerical curvature functions are introduced for noncompact and unbounded polyhedcra in an affine space over an ordered field in this paper. With these functions the generalized Gram-Sommerville and Gauss-Bonnet theorems are obtained and are found to be dual. The Euler characteristic can be interpreted as a zero dimensional curvature measure on the space of polyhedra, and the higher dimensional geometric measures are similarly obtained. The usual numerical curvatures are naturally derived from the non-numerical curvature functions. © 1992.