The Tale of a Geometric Inequality

The path of the rook is a curve γ (possibly, self-intersecting) inside an 8 × 8square. The length of γ should be at least 64 since all squares are visited. Next, I assumed that the rook turned each time by 90 degrees (and never 180). Then the total turn of γ is nπ/2, where n is the number of moves. Thus I traded the original problem for the following one: What is the smallest total turn of a curve of given length inside a given convex domain in the plane? Consider a smooth closed convex plane curve Γ and a smooth, possibly selfintersecting (the technical term is “immersed”) closed curve γ inside it, see Fig.0.1.