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<p>The subject of this article belongs to a “neighborhood” of the four-vertex theorem, which in its simplest form, states that the curvature of a plane oval (a smooth closed curve with positive curvature) has at least four critical points. Since its publication by Syamadas Mukhopadhyaya in 1909, this result and its ramifications have generated a vast literature. We give but one reference: [<CitationRef CitationID="CR5">5</CitationRef>, Lecture 10].</p><p> In what follows, we freely use basic facts of elementary differential geometry of the sphere and the hyperbolic plane, and we omit references to numerous textbooks on the subject.</p>

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A Four-Point Theorem: Yet Another Variation on an Old Theme

  • 期刊论文

  • 发表日期：

    2025年1月25日

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  • Serge Tabachnikov

摘要

The subject of this article belongs to a “neighborhood” of the four-vertex theorem, which in its simplest form, states that the curvature of a plane oval (a smooth closed curve with positive curvature) has at least four critical points. Since its publication by Syamadas Mukhopadhyaya in 1909, this result and its ramifications have generated a vast literature. We give but one reference: [5, Lecture 10].

In what follows, we freely use basic facts of elementary differential geometry of the sphere and the hyperbolic plane, and we omit references to numerous textbooks on the subject.