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

Serge Tabachnikov ()
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Serge Tabachnikov: Pennsylvania State University

The Mathematical Intelligencer, 2025, vol. 47, issue 2, No 13, 175 pages

Abstract: 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.

Date: 2025
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DOI: 10.1007/s00283-024-10399-2

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