Bisubmodular polyhedra, simplicial divisions, and discrete convexity

We consider a class of integer-valued discrete convex functions, called BS-convex functions, defined on integer lattices whose affinity domains are sets of integral points of integral bisubmodular polyhedra. We examine discrete structures of BS-convex functions and give a characterization of BS-convex functions in terms of their convex conjugate functions by means of (discordant) Freudenthal simplicial divisions of the dual space.