On Two Conjectures Concerning Convex Curves

Abstract

Abstract. 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 that i) 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 resultsWe 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 t, cos t, sin 2t, cos 2t, . . . , sin 2kt, cos 2kt) (in some affine coordinates on RP 2k ).Definition. The k-th developable D k (γ) of a curve γ : S 1 → RP n is the union of all k-dimensional osculating subspaces to γ. The hypersurfaceNote that D k (γ) can be considered as the image of the natural associated map γ k :Definition. Let M be a compact manifold of some dimension l ≤ n and φ : M → RP n be a smooth map. By the degree of φ(M ) we understand the supremum of the number of its intersection points withV. SEDYKH AND B. SHAPIRO subspaces for φ(M ) (for example, if dim φ(M ) < dim(M )) then we set the degree of φ(M ) equal to zero. (It is rather obvious that if the Jacobian of φ is nondegenerate at least at one point of M then generic (n − l)-dimensional subspaces exist.)In particular, one can consider the degree of D k (γ) which is positive unless dim D k (γ) < k + 1.

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