  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 d-space is necessarily a d-ball. This is
observed to be true for d at most 3, but false for d=4.
