Two Proofs of a Structural Theorem of Decreasing Minimization on Integrally Convex Sets

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set can be represented as the intersection of a unit discrete cube and a face of the convex hull of the given integrally convex set. The first proof resorts to the Fenchel-type duality theorem in discrete convex analysis and the second is more elementary using Farkas’ lemma.