Convexity and Steinitz's Exchange Property

Abstract

“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. This includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a functionωhas the Steinitz exchange property if and only if it can be extended to a concave functionωsuch that the maximizers of (ω+any linear function) form an integral base polytope. A Frenchel-type min–max theorem and discrete separation theorems are established which imply, as immediate consequences, Frank's discrete separation theorem for submodular functions, Edmonds' intersection theorem, Fujishige's Fenchel-type min–max theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.