Closed cycloids in a normed plane

Abstract

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.