Facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We prove an if-and-only-if version of the Gomory-Johnson Facet Theorem. Finally, we separate the three notions using discontinuous examples.

Automated summary

In this study, three competing notions generalizing the concept of facets of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model are investigated. It is shown that two of the notions, extreme functions and facets, coincide for continuous piecewise linear functions, even without the assumption of rational breakpoints. An if-and-only-if version of the Gomory-Johnson Facet Theorem is proved, and the three notions are further distinguished using discontinuous examples.