Reciprocal domains and Cohen-Macaulay d-complexes in ℝ^d

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in ℝ^d is necessarily a d-ball. This is observed to be true for d ≤ 3, but false for d = 4.