On the total perimeter of disjoint convex bodies

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 $p+2(k-1)d$.