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

TL;DR

This paper presents two distinct proofs of a structural theorem characterizing decreasing minimal elements in integrally convex sets, revealing their geometric structure through duality and elementary methods.

Contribution

It provides two different proofs—one using Fenchel duality and another elementary approach—of a key structural theorem in discrete convex analysis.

Findings

Decreasing minimal elements form an intersection of a cube and a face of the convex hull.

The first proof employs Fenchel-type duality in discrete convex analysis.

The second proof uses Farkas' lemma for an elementary approach.

Abstract

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.