Inclusion and Intersection Relations Between Fundamental Classes of Discrete Convex Functions

TL;DR

This paper provides a comprehensive analysis of the inclusion and intersection relations among various classes of discrete convex functions, highlighting the connections between multimodularity, L-natural-convexity, and M-natural-convexity.

Contribution

It offers a detailed examination of the relationships between fundamental classes of discrete convex functions, clarifying how they include or intersect with each other.

Findings

M-natural-convexity is a subset of integral convexity.

L-natural-convexity and M-natural-convexity together form separable convexity.

Multimodularity's relation to L-natural- and M-natural-convexity is clarified.

Abstract

In discrete convex analysis, various convexity concepts are considered for discrete functions such as separable convexity, L-convexity, M-convexity, integral convexity, and multimodularity. These concepts of discrete convex functions are not mutually independent. For example, M-natural-convexity is a special case of integral convexity, and the combination of L-natural-convexity and M-natural-convexity coincides with separable convexity. This paper aims at a fairly comprehensive analysis of the inclusion and intersection relations for various classes of discrete convex functions. Emphasis is put on the analysis of multimodularity in relation to L-natural-convexity and M-natural-convexity.