Discrete Midpoint Convexity
For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L-(sic)-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at l(infinity)-distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties.