Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra

We relate an axiomatic generalization of matroids, called a jump system, to poly-hedra arising from bisubmodular functions. Unlike the case for usual submodularity, the points of interest are not all the integral points in the relevant polyhedron, but form a subset of them. However, we do show that the convex hull of the set of points of a jump system is a bisubmodular polyhedron, and that the integral points of an integral bisubmodular polyhedron determine a (special) jump system. We also prove addition and composition theorems for jump systems, which have several applications for delta-matroids and matroids.