Self-contracted curves have finite length

Abstract

A curve $\theta$: $I\to E$ in a metric space $E$ equipped with the distance $d$, where $I\subset \R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\{t_i\}_{i=1}^3\subset I$ with $t_1\leq t_2\leq t_3$ one has $d(\theta(t_3),\theta(t_2))\leq d(\theta(t_3),\theta(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\theta$ is bounded, then $\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\cite{Lemenant16sc-rectif}.