  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 edges length
$R$. C_2(R) is equivalent to having constant diameter. We show that smooth
curves satisfying C_n(R) other than the circle do exist for all n, and that the
circle is the only $C^2$ regular curve satisfying C_2(R) and C_4(R') where
$R'=R/\sqrt{2}$. In an addendum, we show that the last assertion holds for any
R and R'. The proofs use only elementary differential calculus and geometry.
