On set functions that can be extended to convex functionals

We give sufficient conditions on the dual generator structures of f, g in order that h is integral when f, g are integral. Using these we derive the (integral) Sandwich Theorem for submodular/supermodular functions and (working with a (0, 1, −1) coefficient matrix generalization of set polyhedra), the 1/2-integral Sandwich Theorem for pseudomatroids. We also study the relative positions of Edmonds Intersection Theorem and Frank's Sandwich Theorem in this class of set functions. It turns out that the former is difficult to generalize unless we generalize the definition of convolution while the latter is routinely generalizable to all pt/dpt functions. Using polyhedral ideas we show that if a set function satisfies the Sandwich Theorem with all supermodular functions it must be submodular.