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.
"A refinement of Foreman’s four-vertex theorem and its dual version." Kyoto J. Math. 52 (4) 743 - 758, Winter 2012. https://doi.org/10.1215/21562261-1728848