WEIGHT-FUNCTIONS, DOUBLE RECIPROCITY LAWS, AND VOLUME FORMULAS FOR LATTICE POLYHEDRA

We extend the concept of manifold with boundary to weight and boundary weight functions, With the new concept, we obtained the double recipr ocity laws for simplicial complexes, cubical complexes, and lattice po lyhedra with weight functions. For a polyhedral manifold with boundary , if the weight function has the constant value 1, then the boundary w eight function has the constant value 1 on the boundary and 0 elsewher e, In particular, for a lattice polyhedral manifold with boundary, our double reciprocity law with a special parameter reduces to the functi onal equation of Macdonald; for a lattice polytope especially, the dou ble reciprocity law with a special parameter reduces to the reciprocit y law of Ehrhart. Several volume formulas for lattice polyhedra are ob tained from the properties of the double reciprocity law Moreover, the idea of weight and boundary weight leads to a new homology that is no t homotopy invariant, but only homeomorphic invariant.