THE INCIDENCE ALGEBRA OF POLYHEDRA OVER THE MINKOWSKI ALGEBRA

The Minkowski algebra of polyhedra over an ordered field is introduced , and the incidence algebra of polyhedra over this algebra is consider ed. The Gram-Sommerville and Gauss-Bonnet theorems and their analogs f or relatively open convex polyhedra are given by various Mobius invers ion formulas. Cheeger's Chern-Gauss-Bonnet density formula for piecewi se flat spaces is generalized to non-compact and unbounded polyhedra, and its non-numerical forms are naturally expressed in terms of interi or cone chains. The tangential densities and interior cone functions a re defined as the dual versions of the Chern-Gauss-Bonnet densities an d exterior cone functions. The tangential density formulas and their n on-numerical forms are similarly represented in terms of exterior curv atures and exterior cone chains. (C) 1996 Academic Press, Inc.