On S.L. Tabachnikov's conjecture

Abstract

S. L. Tabachnikov's conjecture is proved: for any closed curve $\Gamma$ lying inside convex closed curve ${\Gamma }_1$ the mean absolute curvature $T(\Gamma )$ exceeds $T({\Gamma }_1)$ if $\Gamma \ne k{\Gamma }_1$. An inequality $T(\Gamma )\ge T({\Gamma }_1)$ is proved for curves in a hemisphere.