Note on the polyhedral description of the Minkowski sum of two L-convex sets

Abstract

AbstractL-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L$$_{2}$$ 2 -convex sets, is an intriguing object that is closely related to polymatroid intersection. This paper reveals the polyhedral description of an L$$_{2}$$ 2 -convex set, together with the observation that the convex hull of an L$$_{2}$$ 2 -convex set is a box-TDI polyhedron. Two different proofs are given for the polyhedral description. The first is a structural short proof, relying on the conjugacy theorem in discrete convex analysis, and the second is a direct algebraic proof, based on Fourier–Motzkin elimination. The obtained results admit natural graph representations. Implications of the obtained results in discrete convex analysis are also discussed.