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
A strictly convex curve is a -regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve , and take two distinct tangent lines and of . A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of , which are tangent to and , as a corollary of Ghys’s theorem on diffeomorphisms of . 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 . We also show that the dual version of Foreman’s four-vertex theorem is almost equivalent to the Ghys’s theorem.
Kyoto J. Math., Volume 52, Number 4 (2012), 743-758.
First available in Project Euclid: 15 November 2012
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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