## Kyoto Journal of Mathematics

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

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

Gudlaugur Thorbergsson and Masaaki Umehara

#### 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

**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