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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.

Un cône ouvert proprement convexe C de ℝm est dit divisible si il existe un sousgroupe discret Γ de GL(ℝm) qui préserve C et tel que quotient Γ\C est compact. Nous décrivons l'adhérence de Zariski d'un tel groupe Γ. ¶ Nous montrons que si C n'est ni un produit ni un cône symétrique alors Γ est Zariski dense dans GL(ℝm).

Determinant is a most useful tool in every branch of mathematics, especially in linear mathematics. What kind of quantities do take a similar universal important role as determinant, in advanced branches of mathematics? In the present articles, showing the usefulness of semi-invariants in the classical invariant theory, we shall give a partial answer of the above question.