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One easily proves that two disjoint closed disks in \(\mathbb R^2\) are separated by a line. With relatively little extra effort one can prove that two disjoint closed convex sets in \(\mathbb R^3\) are separated by a plane, and generalize this to any two disjoint convex closed sets in \(\mathbb R^n\) . However, this is not what we need for the proof of Farkas’ lemma, a key tool for the proof of the probabilistic consistency theorem in the next chapter. Here, one of the two disjoint convex sets to be separated is not closed.

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Minkowski Hyperplane Separation, Farkas’ Lemma

  • 章节

  • 发表日期：

    2026年

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  • Daniele Mundici

摘要

One easily proves that two disjoint closed disks in \(\mathbb R^2\) are separated by a line. With relatively little extra effort one can prove that two disjoint closed convex sets in \(\mathbb R^3\) are separated by a plane, and generalize this to any two disjoint convex closed sets in \(\mathbb R^n\) . However, this is not what we need for the proof of Farkas’ lemma, a key tool for the proof of the probabilistic consistency theorem in the next chapter. Here, one of the two disjoint convex sets to be separated is not closed.