In this note we introduce a pseudometric on convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a short proof of the theorem by Pinchasi that the sum of perimeters of k convex planar bodies with disjoint interiors contained in a convex body of perimeter p and diameter d is not greater than p+2(k−1)d.
Akopyan, Arseniy, and Alexey Glazyrin. "On the total perimeter of disjoint convex bodies." arXiv preprint arXiv:2308.16340 (2023).