On Tabachnikov’s Conjecture

Tabachnikov’s conjecture is proved: for any closed curve Γ lying inside a convex closed curve Γ1 the mean absolute curvature T (Γ) exceeds T (Γ1) if Γ = kΓ1. §1. The problem setting and main ideas Let Γ(s), s ∈ [0, L(Γ)], be a naturally parametrized closed curve on a plane. We say that Γ(s) belongs to the class BV 1 if the velocity Γ′(s) exists and is continuous everywhere on [0, L(Γ)] except for a countable set; at the points of this set Γ′ has left and right limits, and the variation of Γ′ is bounded. The full variation of Γ′ is called the full rotation of the curve Γ and is denoted by V (Γ). The full rotation has the following properties. 1◦. For C-smooth curves, the full rotation is equal to the integral of the absolute value of the curvature with respect to arc length. 2◦. The full rotation of a closed polygonal line equals the sum of the external angles at all of its vertices. 3◦. The full rotation of a closed convex curve exists and is equal to 2π. We define the mean absolute curvature of a curve Γ ∈ BV 1 as its full rotation divided by its length: T (Γ) = V (Γ)/L(Γ). In [1], S. L. Tabachnikov formulated the following conjecture, which he called the DNA inequality. Theorem P. 1. The mean absolute curvature T (Γ) of a closed curve Γ ∈ BV 1 (“DNA”) lying inside a closed convex curve Γ1 (“cell”) does not exceed T (Γ1). 2. If T (Γ) = T (Γ1), then Γ is a multiple circuit of Γ1. A survey of results concerning this conjecture and its generalisations was given in [1]. The first part of Theorem P was proved in [2]. We prove the DNA inequality in full generality. The proof of the first part partially follows the strategy of [2], but is clearer and is used in the proof of the second part. In order to make the paper self-contained, we give (significantly simpler) proofs of all lemmas from [2] that we use. Without loss of generality we may assume that Γ1 is the boundary of the convex hull of Γ. 2000 Mathematics Subject Classification. Primary 53A04; Secondary 52A40, 52A10.