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Winter 2012 A refinement of Foreman’s four-vertex theorem and its dual version
Gudlaugur Thorbergsson, Masaaki Umehara
Kyoto J. Math. 52(4): 743-758 (Winter 2012). DOI: 10.1215/21562261-1728848


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.


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Gudlaugur Thorbergsson. Masaaki Umehara. "A refinement of Foreman’s four-vertex theorem and its dual version." Kyoto J. Math. 52 (4) 743 - 758, Winter 2012.


Published: Winter 2012
First available in Project Euclid: 15 November 2012

zbMATH: 1262.53004
MathSciNet: MR2998909
Digital Object Identifier: 10.1215/21562261-1728848

Primary: 53A04 , 53A20 , 53C75

Rights: Copyright © 2012 Kyoto University


Vol.52 • No. 4 • Winter 2012
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