Natural pseudo-distances between closed curves

Abstract

Let us consider two closed curves $mathcal{M}$, $mathcal{N}$ of class $C^1$ and two functions $varphi:{mathcal{M}}rightarrow R$, $psi:mathcal{N}rightarrow R$ of class $C^1$, called measuring functions. The natural pseudo-distance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. The problem of finding the possible values for $d$ naturally arises.\nIn this paper we prove that under appropriate hypotheses the\nnatural pseudo-distance equals either $|c_1-c_2|$ or\n$frac{1}{2}|c_1-c_2|$, where $c_1$ and $c_2$ are two suitable\ncritical values of the measuring functions. This equality shows\nthat the relations between the natural pseudo-distance and the\ncritical values of the measuring functions previously obtained in\nhigher dimensions can be made stronger in the particular case of\nclosed curves. Moreover, the examples we give in this paper show\nthat our result cannot be further improved, and therefore it\ncompletely solves the problem of determining the possible values\nfor $d$ in the $1$-dimensional case.