Gross substitution, discrete convexity, and submodularity

We consider a class of functions satisfying the gross-substitutes property (GS-functions). We show that GS-functions are concave functions, whose parquets are constituted by quasipolymatroids. 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, form a proper subclass of the class of GS-functions. We provide an additional characterization of PM-functions. ? 2003 Elsevier B.V. All rights reserved.