Curves of constant diameter and inscribed polygons

A simple closed curve in the Euclidean plane is said to have property(C_n(R)) if at each point we can inscribe a unique regular $n$-gon with edgeslength $R$. (C_2(R)) is equivalent to having constant diameter. We show thatsmooth curves satisfying (C_n(R)) other than the circle do exist for all n, andthat the circle is the only $C^2$ regular curve satisfying (C_2(R)) and(C_4(R')) where .$R'=R/\sqrt{2}$. The proofs use only elementary differentialgeometry (curvature,etc).