Convexity and Steinitz's exchange property

A theory of “convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids, and separable concave functions on the integral base polytope. It is shown that a function ω has the exchange property if and only if it can be extended to a concave function \(\bar \omega\)such that the maximizers of (\(\bar \omega\)+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are given, which contain, e.g., Frank's discrete separation theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.