## Kyoto Journal of Mathematics

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

#### Abstract

A strictly convex curve is a $C^{\infty}$-regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve $\Gamma$, and take two distinct tangent lines $l_{1}$ and $l_{2}$ of $\Gamma$. A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of $\Gamma$, which are tangent to $l_{1}$ and $l_{2}$, as a corollary of Ghys’s theorem on diffeomorphisms of $S^{1}$. 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 $\Gamma$. 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

#### 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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