Some Geometric Properties of Closed Space Curves and Convex Bodies

The main results of the paper are as follows. 1. On each smooth closed oriented curve in Rn, there exist two points the oriented tangents at which form an angle greater than π/2 + sin−1 1 n−1 . 2. If n is odd, then an (n+1)-gon with equal sides and lying in a hyperplane can be inscribed in each smooth closed Jordan curve in Rn. In particular, a rhombus can be inscribed in each closed curve in R3. 3. A right prism with rhombic base and an arbitrary ratio of the base edge to the lateral edge can be inscribed in each smooth strictly convex body K ⊂ R3. In the sequel, by a convex body K ⊂ R we mean a compact convex body with nonempty interior. A convex body K is said to be strictly convex if it has a unique intersection point with each of its support planes. As usual, for A ⊂ R, we denote by conv(A) and by int(A) the convex hull and the interior of A, respectively. As usual, by G2 (R ) (respectively, by G2(R)) we denote the Grassmann manifold of oriented (respectively, nonoriented) planes passing through the origin in R, and by E (+) 2 (R ) → G 2 (R) the tautological vector bundle over G (+) 2 (R ) in which the fiber over a plane g ∈ G 2 (R) is the plane itself regarded as a vector subspace of R. §1. Oriented angles between oriented tangents to a space curve Throughout the paper, γ(t) : [0, 1] R is a regularly parametrized curve. We orient the tangents to γ in accordance with the orientation of γ. We begin with the following obvious statement. Lemma. Let a ∈ R be a nonzero vector. Then either γ has oriented tangents that form acute as well as obtuse angles with a, or γ lies in the hyperplane orthogonal to a. Indeed, multiplying the equation ∫ 1 0 γ′ dt = 0 by a, we obtain ∫ 1 0 γ′ ·a dt = 0; therefore, the function under the integral either takes both positive and negative values, or is zero identically. Theorem 1. 1. For n ≥ 3, each curve γ : [0, 1] R has two oriented tangents that form an angle greater than π/2 + sin−1 1 n−1 . 2. Any two immersed curves γ1, γ2 : [0, 1] R have oriented tangents that form acute and obtuse angles, respectively. Both estimates are best possible. 2000 Mathematics Subject Classification. Primary 51H99.