Discrete 2-convex functions

We focus on a new class of integrally convex functions which we call discrete 2-convex functions. Discrete 2-convexity generalizes known classes of integrally convex functions such as the well-established M-/M♮-convex and L-/L♮-convex functions by Murota et al., the recently investigated globally/locally discrete midpoint convex functions by Moriguchi, Murota, Tamura, and Tardella, the directed discrete midpoint convex functions by Tamura and Tsurumi, and BS∗-convex and UJ-convex functions by one of the authors. We provide a unifying view of all these functions within the class of integrally convex functions having discrete 2-convexity. We also introduce a new subclass of discrete 2-convex functions, called signed discrete 2-convex functions and we consider signed discrete 2-convex functions with a locally hereditary orientation property. We show that parallelogram inequalities, scalability, and proximity hold for such signed discrete 2-convex functions, which include globally/locally discrete midpoint convex functions and directed discrete midpoint convex functions. Hence, our results extend similar results recently established by Moriguchi, Murota, Tamura, and Tardella and by Tamura and Tsurumi.