Symmetrization of Convex Planar Curves

Given a closed convex planar curve, we call great chords the segment connecting two points with parallel tangents. We call great diagonals the support lines of the great chords and mid-parallels the lines through the mid-point of a great chord parallel to the corresponding tangents. Two curves are called parallel if the corresponding great diagonals are parallel. In this paper, we define the parallel diagonal (PD) transform of a convex curve γ as a convex curve δ whose great diagonals coincide with the mid-parallels of γ and whose mid-parallels are parallel to the great diagonals of γ. Applying twice the PD transform, we obtain a transformation S that preserves parallelism of the curves. The main result of the paper says that the sequence of iterations S(γ) converges uniformly to a symmetric curve parallel to γ. Mathematics Subject Classification (2010). 53A10.