ar X iv : m at h / 02 08 21 8 v 1 [ m at h . A G ] 2 8 A ug 2 00 2 TWO CONJECTURES ON CONVEX CURVES

In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show i) that the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them. The question (discussed in [EG1] and [So4]) whether the second conjecture is true in the special case of rational normal curves still remains open. §1. Introduction and results We start with some important notions. Main definition. A smooth closed curve γ : S 1 → RP n is called locally convex if the local multiplicity of intersection of γ with any hyperplane H ⊂ RP n at any of the intersection points does not exceed n = dim RP n and globally convex or just convex if the above condition holds for the global multiplicity, i.e for the sum of local multiplicities. Local convexity of γ is a simple requirement of nondegeneracy of the osculating Frenet n-frame of γ, i.e. of the linear independence of γ ′ (t), ..., γ n (t) for any t ∈ S 1. Global convexity is a nontrivial property equivalent to the fact that the (n + 1)-tuple of γ's homogeneous coordinates forms a Tschebychev system of functions, see e.g. [KS]. The simplest examples of convex curves are the rational normal curve ρ n : t → (t, t 2 ,. .. , t n) (in some affine coordinates on RP n) and the standard trigonometric curve τ 2k : t → sin 2kt, cos 2kt) (in some affine coordinates on RP 2k).