Approximation of Minimal Functions by Extreme Functions

TL;DR

This paper proves that in the n-dimensional Gomory-Johnson model, every continuous minimal function can be closely approximated by an extreme function, extending previous 1-dimensional results to higher dimensions.

Contribution

It establishes that for all dimensions n ≥ 1, continuous minimal functions can be approximated arbitrarily well by extreme functions, solving an open problem.

Findings

Every continuous minimal function in n dimensions can be approximated by an extreme function.

Extension of 1D approximation results to higher dimensions.

Confirms the density of extreme functions in the set of minimal functions.

Abstract

In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional version of this result was left as an open question. In the present paper, we settle this query in the affirmative: for any integer $n \geq 1$, every continuous minimal function can be arbitrarily well approximated by an extreme function in the $n$-dimensional Gomory-Johnson model.