Gross Substitution, Discrete Convexity and Submodularity

We consider a class of functions satisfying the gross-substitution property (GS-functions). We show that GS-functions are concave functions whose parquets are constributed by quasi-polymatroids. The class of conjugate functions to GS-functions turns out to be the class of polyhedral supermodular functions.The class of polyhedral GS-functions is a proper subclass of the class of polyhedral submodular functions. PM-functions, concave functions whose parquets are constituted by g-polymatroids, from a proper subclass of the class of GS-function. We provide an additional characterization of PM-functions.