Discrete Fenchel duality for a pair of integrally convex and separable convex functions

Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min–max theorem for a pair of integer-valued M^♮-convex functions generalizes the min–max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min–max formula for a pair of integer-valued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M^♮-convex functions and L^♮-convex functions, whereas separable convex functions are characterized as those functions which are both M^♮-convex and L^♮-convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier–Motzkin elimination.