  Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar
curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns
out that the radius of curvature and the support function of a
$\mathcal{P}$-cycloid satisfy a differential equation of Sturm-Liouville type.
By studying this equation we can describe all closed hypocycloids and
epicycloids with a given number of cusps. We can also find an orthonormal basis
of ${\mathcal C}^0(S^1)$ with a natural decomposition into symmetric and
anti-symmetric functions, which are support functions of symmetric and constant
width curves, respectively. As applications, we prove that the iterations of
involutes of a closed curve converge to a constant and a generalization of the
Sturm-Hurwitz Theorem. We also prove versions of the four vertices theorem for
closed curves and six vertices theorem for closed constant width curves.
