Kyoto Journal of Mathematics

A refinement of Foreman’s four-vertex theorem and its dual version

Gudlaugur Thorbergsson and Masaaki Umehara

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Abstract

A strictly convex curve is a C-regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve Γ, and take two distinct tangent lines l1 and l2 of Γ. A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of Γ, which are tangent to l1 and l2, as a corollary of Ghys’s theorem on diffeomorphisms of S1. In this paper, we prove a refinement of Foreman’s result. We then prove a projectively dual version of our refinement, which is a claim about semiosculating conics passing through two fixed points on Γ. We also show that the dual version of Foreman’s four-vertex theorem is almost equivalent to the Ghys’s theorem.

Article information

Source
Kyoto J. Math., Volume 52, Number 4 (2012), 743-758.

Dates
First available in Project Euclid: 15 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1352987532

Digital Object Identifier
doi:10.1215/21562261-1728848

Mathematical Reviews number (MathSciNet)
MR2998909

Zentralblatt MATH identifier
1262.53004

Subjects
Primary: 53A04: Curves in Euclidean space 53A20: Projective differential geometry 53C75: Geometric orders, order geometry [See also 51Lxx]

Citation

Thorbergsson, Gudlaugur; Umehara, Masaaki. A refinement of Foreman’s four-vertex theorem and its dual version. Kyoto J. Math. 52 (2012), no. 4, 743--758. doi:10.1215/21562261-1728848. https://projecteuclid.org/euclid.kjm/1352987532


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References

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