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.

中文翻译：

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Gomory-Johnson 无限群问题的面、弱面和极值函数

我们研究了三个相互竞争的概念，这些概念将有限维多面体的一个方面的概念推广到无限维 Gomory-Johnson 模型。已知这些概念对于具有有理断点的连续分段线性函数是一致的。我们展示了两个概念，极值函数和面，对于连续分段线性函数的情况是一致的，消除了关于有理断点的假设。我们证明了 Gomory-Johnson Facet Theorem 的当且仅当版本。最后，我们使用不连续的例子将这三个概念分开。