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

Abstract

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.