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.