Odd Submodular Functions, Dilworth Functions and Discrete Convex Functions

Three set-function classes more general than submodular ones are discussed. An odd submodular function defines a box totally dual integral system. An integral odd submodular function and its Dilworth truncation, i.e., a Dilworth function, give rise to the same polyhedron with integral vertices. The minimum of two submodular functions is an odd submodular function. The convolution of two submodular functions is a Dilworth function. A Dilworth function is a discrete convex function. A discrete convex function can be characterized by its convex hull, subgradients, epigraph and general subadditivity. Discrete convexity is preserved under many natural operations.