  The closure of the set of entropy functions associated with n discrete
variables, Gammar*n, is a convex cone in (2n-1)- dimensional space, but its
full characterization remains an open problem. In this paper, we map Gammar*n
to an n-dimensional region Phi*n by averaging the joint entropies with the same
number of variables, and show that the simpler Phi*n can be characterized
solely by the Shannon-type information inequalities
