Curves of constant diameter and inscribed polygons

If one drops the unicity assumption, (C(D)) is the property of having constant diameter, which is in fact equivalent (for closed curves in the plane) to having constant width or constant breadth (for the definitions and the proof of the equivalence, see [8, chap. 25]). It is a surprise to many (including me !) that curves of constant diameter other that the circle do exist. The simplest examples (due to Reuleaux [9]) are pictured below. They are built with circle arcs whose centers are marked with a black dot. Notice that these curves are not C2.