Recent progress on integrally convex functions

Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on integrally convex functions with some new technical results. Topics covered in this paper include characterizations of integral convex sets and functions, operations on integral convex sets and functions, optimality criteria for minimization with a proximity-scaling algorithm, integral biconjugacy, and the discrete Fenchel duality. While the theory of M-convex and L-convex functions has been built upon fundamental results on matroids and submodular functions, developing the theory of integrally convex functions requires more general and basic tools such as the Fourier-Motzkin elimination.

Automated summary

This paper provides a comprehensive survey of integrally convex functions and presents some new technical results. The topics covered include characterizations, operations, optimality criteria, integral biconjugacy, and discrete Fenchel duality. The development of the theory of integrally convex functions requires more general and basic tools such as the Fourier-Motzkin elimination compared to the theory of M-convex and L-convex functions.