Scaling and proximity properties of integrally convex functions

In discrete convex analysis, the scaling and proximity properties for the class of L^♮-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L^♮-convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L^♮-convex functions.